Recursive Derivative Orders, Exact Polynomial Coefficients and Existence Boundaries
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batch-20261014-advance-20261111-r02-higher-order-derivative-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[1, 2, 3, 4] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(3, 0, Yes, f_zero_equals_f, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (3, 0, Yes, f_zero_equals_f, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
2. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[2, 4, 6, 8, 10] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(4, 1, Yes, recursive_derivative_chain_required, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (4, 1, Yes, recursive_derivative_chain_required, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
3. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[3, 6, 9, 12, 15, 18] for f(t)=sum coefficients[k]*t^k and derivative order=2≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(5, 2, Yes, recursive_derivative_chain_required, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (5, 2, Yes, recursive_derivative_chain_required, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
4. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[4, 8, 12, 16, 20, 24, 28] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(6, 3, Yes, recursive_derivative_chain_required, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (6, 3, Yes, recursive_derivative_chain_required, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
5. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[5, 10, 15, 20] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(3, 4, Yes, recursive_derivative_chain_required, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (3, 4, Yes, recursive_derivative_chain_required, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
6. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[6, 12, 18, 24, 30] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(4, 5, Yes, recursive_derivative_chain_required, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (4, 5, Yes, recursive_derivative_chain_required, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
7. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[7, 14, 21, 28, 35, 42] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(5, 6, Yes, recursive_derivative_chain_required, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (5, 6, Yes, recursive_derivative_chain_required, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
8. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[8, 16, 24, 32, 40, 48, 56] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(6, 7, Yes, recursive_derivative_chain_required, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (6, 7, Yes, recursive_derivative_chain_required, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
9. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[9, 18, 27, 36] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(3, 3, Yes, recursive_derivative_chain_required, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (3, 3, Yes, recursive_derivative_chain_required, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
10. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[10, 20, 30, 40, 50] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(4, 3, Yes, recursive_derivative_chain_required, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (4, 3, Yes, recursive_derivative_chain_required, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
11. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[11, 22, 33, 44, 55, 66] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(5, 3, Yes, recursive_derivative_chain_required, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (5, 3, Yes, recursive_derivative_chain_required, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
12. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[12, 24, 36, 48, 60, 72, 84] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(6, 3, Yes, recursive_derivative_chain_required, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (6, 3, Yes, recursive_derivative_chain_required, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
13. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[13, 26, 39, 52] for f(t)=sum coefficients[k]*t^k and derivative order=2≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(3, 2, Yes, recursive_derivative_chain_required, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (3, 2, Yes, recursive_derivative_chain_required, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
14. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[14, 28, 42, 56, 70] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(4, 1, Yes, recursive_derivative_chain_required, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (4, 1, Yes, recursive_derivative_chain_required, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
15. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[15, 30, 45, 60, 75, 90] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(5, 0, Yes, f_zero_equals_f, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (5, 0, Yes, f_zero_equals_f, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
16. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[16, 32, 48, 64, 80, 96, 112] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(6, 7, Yes, recursive_derivative_chain_required, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (6, 7, Yes, recursive_derivative_chain_required, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
17. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[17, 34, 51, 68] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(3, 1, Yes, recursive_derivative_chain_required, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (3, 1, Yes, recursive_derivative_chain_required, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
18. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[18, 36, 54, 72, 90] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(4, 5, Yes, recursive_derivative_chain_required, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (4, 5, Yes, recursive_derivative_chain_required, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
19. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[19, 38, 57, 76, 95, 114] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(5, 4, Yes, recursive_derivative_chain_required, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (5, 4, Yes, recursive_derivative_chain_required, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
20. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[20, 40, 60, 80, 100, 120, 140] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(6, 3, Yes, recursive_derivative_chain_required, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (6, 3, Yes, recursive_derivative_chain_required, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
21. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[21, 42, 63, 84] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(3, 0, Yes, f_zero_equals_f, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (3, 0, Yes, f_zero_equals_f, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
22. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[22, 44, 66, 88, 110] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(4, 3, Yes, recursive_derivative_chain_required, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (4, 3, Yes, recursive_derivative_chain_required, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
23. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[23, 46, 69, 92, 115, 138] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(5, 1, Yes, recursive_derivative_chain_required, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (5, 1, Yes, recursive_derivative_chain_required, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
24. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[24, 48, 72, 96, 120, 144, 168] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(6, 7, Yes, recursive_derivative_chain_required, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (6, 7, Yes, recursive_derivative_chain_required, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
25. Order Zero Is the Original Function; Later Orders Need the Recursive Chain
Use original low-to-high coefficients=[25, 50, 75, 100] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Give (declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate).
declared_degree_bound is the original coefficient-list length minus1. With trailing zero coefficients it is an upper bound, not the minimal degree after trimming.
View answer and explanation
(3, 4, Yes, recursive_derivative_chain_required, Yes)
Keep the declared degree bound, natural order and zero-order convention. A polynomial supplies every finite derivative in this family, but existence of an n-th derivative for an arbitrary function does not imply its next derivative or analyticity. The exact result, retaining the requested order, is (3, 4, Yes, recursive_derivative_chain_required, Yes).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-order-qualification · candidate
26. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[1, 2, 3, 4] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([1, 2, 3, 4], 0, [1, 2, 3, 4], -2, -23, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([1, 2, 3, 4], 0, [1, 2, 3, 4], -2, -23, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
27. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[2, 4, 6, 8, 10] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([2, 4, 6, 8, 10], 1, [4, 12, 24, 40], -1, -24, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([2, 4, 6, 8, 10], 1, [4, 12, 24, 40], -1, -24, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
28. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[3, 6, 9, 12, 15, 18] for f(t)=sum coefficients[k]*t^k and derivative order=2≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([3, 6, 9, 12, 15, 18], 2, [18, 72, 180, 360], 0, 18, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([3, 6, 9, 12, 15, 18], 2, [18, 72, 180, 360], 0, 18, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
29. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[4, 8, 12, 16, 20, 24, 28] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([4, 8, 12, 16, 20, 24, 28], 3, [96, 480, 1440, 3360], 1, 5376, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([4, 8, 12, 16, 20, 24, 28], 3, [96, 480, 1440, 3360], 1, 5376, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
30. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[5, 10, 15, 20] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([5, 10, 15, 20], 4, [0], 2, 0, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([5, 10, 15, 20], 4, [0], 2, 0, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
31. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[6, 12, 18, 24, 30] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([6, 12, 18, 24, 30], 5, [0], -2, 0, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([6, 12, 18, 24, 30], 5, [0], -2, 0, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
32. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[7, 14, 21, 28, 35, 42] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([7, 14, 21, 28, 35, 42], 6, [0], -1, 0, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([7, 14, 21, 28, 35, 42], 6, [0], -1, 0, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
33. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[8, 16, 24, 32, 40, 48, 56] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([8, 16, 24, 32, 40, 48, 56], 7, [0], 0, 0, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([8, 16, 24, 32, 40, 48, 56], 7, [0], 0, 0, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
34. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[9, 18, 27, 36] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([9, 18, 27, 36], 3, [216], 1, 216, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([9, 18, 27, 36], 3, [216], 1, 216, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
35. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[10, 20, 30, 40, 50] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([10, 20, 30, 40, 50], 3, [240, 1200], 2, 2640, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([10, 20, 30, 40, 50], 3, [240, 1200], 2, 2640, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
36. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[11, 22, 33, 44, 55, 66] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([11, 22, 33, 44, 55, 66], 3, [264, 1320, 3960], -2, 13464, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([11, 22, 33, 44, 55, 66], 3, [264, 1320, 3960], -2, 13464, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
37. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[12, 24, 36, 48, 60, 72, 84] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([12, 24, 36, 48, 60, 72, 84], 3, [288, 1440, 4320, 10080], -1, -6912, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([12, 24, 36, 48, 60, 72, 84], 3, [288, 1440, 4320, 10080], -1, -6912, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
38. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[13, 26, 39, 52] for f(t)=sum coefficients[k]*t^k and derivative order=2≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([13, 26, 39, 52], 2, [78, 312], 0, 78, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([13, 26, 39, 52], 2, [78, 312], 0, 78, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
39. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[14, 28, 42, 56, 70] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([14, 28, 42, 56, 70], 1, [28, 84, 168, 280], 1, 560, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([14, 28, 42, 56, 70], 1, [28, 84, 168, 280], 1, 560, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
40. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[15, 30, 45, 60, 75, 90] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([15, 30, 45, 60, 75, 90], 0, [15, 30, 45, 60, 75, 90], 2, 4815, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([15, 30, 45, 60, 75, 90], 0, [15, 30, 45, 60, 75, 90], 2, 4815, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
41. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[16, 32, 48, 64, 80, 96, 112] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([16, 32, 48, 64, 80, 96, 112], 7, [0], -2, 0, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([16, 32, 48, 64, 80, 96, 112], 7, [0], -2, 0, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
42. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[17, 34, 51, 68] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([17, 34, 51, 68], 1, [34, 102, 204], -1, 136, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([17, 34, 51, 68], 1, [34, 102, 204], -1, 136, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
43. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[18, 36, 54, 72, 90] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([18, 36, 54, 72, 90], 5, [0], 0, 0, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([18, 36, 54, 72, 90], 5, [0], 0, 0, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
44. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[19, 38, 57, 76, 95, 114] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([19, 38, 57, 76, 95, 114], 4, [2280, 13680], 1, 15960, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([19, 38, 57, 76, 95, 114], 4, [2280, 13680], 1, 15960, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
45. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[20, 40, 60, 80, 100, 120, 140] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([20, 40, 60, 80, 100, 120, 140], 3, [480, 2400, 7200, 16800], 2, 168480, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([20, 40, 60, 80, 100, 120, 140], 3, [480, 2400, 7200, 16800], 2, 168480, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
46. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[21, 42, 63, 84] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([21, 42, 63, 84], 0, [21, 42, 63, 84], -2, -483, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([21, 42, 63, 84], 0, [21, 42, 63, 84], -2, -483, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
47. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[22, 44, 66, 88, 110] for f(t)=sum coefficients[k]*t^k and derivative order=3≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([22, 44, 66, 88, 110], 3, [528, 2640], -1, -2112, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([22, 44, 66, 88, 110], 3, [528, 2640], -1, -2112, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
48. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[23, 46, 69, 92, 115, 138] for f(t)=sum coefficients[k]*t^k and derivative order=1≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([23, 46, 69, 92, 115, 138], 1, [46, 138, 276, 460, 690], 0, 46, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([23, 46, 69, 92, 115, 138], 1, [46, 138, 276, 460, 690], 0, 46, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
49. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[24, 48, 72, 96, 120, 144, 168] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([24, 48, 72, 96, 120, 144, 168], 7, [0], 1, 0, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([24, 48, 72, 96, 120, 144, 168], 7, [0], 1, 0, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
50. Exact Falling Factorials Preserve Low-to-High Derivative Coefficients
Use original low-to-high coefficients=[25, 50, 75, 100] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (coefficients, order, derivative_coefficients, point, value, certificate).
View answer and explanation
([25, 50, 75, 100], 4, [0], 2, 0, falling_factorial_coefficients_exact)
Differentiate each coefficient with its falling-factorial multiplier, retain the original coefficient order and evaluate exactly at the declared integer point. Order0 retains the original list; after all degrees are exhausted the derivative list is [0]. Unsafe intermediate Number arithmetic is rejected before projection. The exact result, retaining the requested order, is ([25, 50, 75, 100], 4, [0], 2, 0, falling_factorial_coefficients_exact).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-polynomial-nth-derivative · candidate
51. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[1, 2, 3, 4] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(3, 4, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (3, 4, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
52. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[2, 4, 6, 8, 10] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(4, 6, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (4, 6, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
53. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[3, 6, 9, 12, 15, 18] for f(t)=sum coefficients[k]*t^k and derivative order=8≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(5, 8, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (5, 8, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
54. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[4, 8, 12, 16, 20, 24, 28] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(6, 7, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (6, 7, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
55. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[5, 10, 15, 20] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(3, 5, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (3, 5, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
56. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[6, 12, 18, 24, 30] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(4, 7, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (4, 7, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
57. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[7, 14, 21, 28, 35, 42] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(5, 6, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (5, 6, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
58. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[8, 16, 24, 32, 40, 48, 56] for f(t)=sum coefficients[k]*t^k and derivative order=8≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(6, 8, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (6, 8, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
59. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[9, 18, 27, 36] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(3, 6, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (3, 6, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
60. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[10, 20, 30, 40, 50] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(4, 5, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (4, 5, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
61. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[11, 22, 33, 44, 55, 66] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(5, 7, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (5, 7, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
62. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[12, 24, 36, 48, 60, 72, 84] for f(t)=sum coefficients[k]*t^k and derivative order=9≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(6, 9, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (6, 9, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
63. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[13, 26, 39, 52] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(3, 4, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (3, 4, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
64. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[14, 28, 42, 56, 70] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(4, 6, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (4, 6, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
65. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[15, 30, 45, 60, 75, 90] for f(t)=sum coefficients[k]*t^k and derivative order=8≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(5, 8, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (5, 8, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
66. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[16, 32, 48, 64, 80, 96, 112] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(6, 7, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (6, 7, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
67. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[17, 34, 51, 68] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(3, 5, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (3, 5, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
68. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[18, 36, 54, 72, 90] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(4, 7, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (4, 7, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
69. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[19, 38, 57, 76, 95, 114] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(5, 6, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (5, 6, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
70. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[20, 40, 60, 80, 100, 120, 140] for f(t)=sum coefficients[k]*t^k and derivative order=8≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(6, 8, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (6, 8, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
71. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[21, 42, 63, 84] for f(t)=sum coefficients[k]*t^k and derivative order=6≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(3, 6, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (3, 6, [0], -2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
72. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[22, 44, 66, 88, 110] for f(t)=sum coefficients[k]*t^k and derivative order=5≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(4, 5, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (4, 5, [0], -1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
73. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[23, 46, 69, 92, 115, 138] for f(t)=sum coefficients[k]*t^k and derivative order=7≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=0. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(5, 7, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (5, 7, [0], 0, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
74. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[24, 48, 72, 96, 120, 144, 168] for f(t)=sum coefficients[k]*t^k and derivative order=9≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=1. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(6, 9, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (6, 9, [0], 1, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
75. Orders above the Declared Polynomial Degree Bound Give Zero
Use original low-to-high coefficients=[25, 50, 75, 100] for f(t)=sum coefficients[k]*t^k and derivative order=4≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=2. Give (declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate).
View answer and explanation
(3, 4, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim)
Require order strictly above the coefficient-list degree bound, giving [0] and value0. This sufficient bound remains valid with trailing zero coefficients; it is not a general analyticity or Taylor-series convergence claim. The exact result, retaining the requested order, is (3, 4, [0], 2, 0, Yes, orders_above_polynomial_degree_are_zero_without_analyticity_claim).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-zero-beyond-degree · candidate
76. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=1 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(1, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (1, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
77. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=2 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(2, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (2, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
78. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=3 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(3, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (3, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
79. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=4 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(4, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (4, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
80. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=5 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(5, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (5, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
81. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=6 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(6, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (6, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
82. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=7 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(7, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (7, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
83. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=8 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(8, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (8, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
84. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=9 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(9, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (9, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
85. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=10 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(10, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (10, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
86. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=11 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(11, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (11, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
87. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=12 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(12, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (12, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
88. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=13 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(13, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (13, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
89. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=14 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(14, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (14, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
90. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=15 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(15, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (15, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
91. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=16 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(16, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (16, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
92. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=17 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(17, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (17, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
93. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=18 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(18, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (18, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
94. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=19 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(19, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (19, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
95. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=20 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(20, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (20, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
96. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=21 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(21, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (21, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
97. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=22 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(22, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (22, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
98. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=23 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(23, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (23, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
99. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=24 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(24, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (24, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
100. A First Derivative May Exist Where the Second Does Not
Use nonzero scale=25 and the whole real function f(x)=scale*x*abs(x), at x=0. Give (scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary).
View answer and explanation
(25, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits)
The first derivative at0 exists and equals0. Away from0 the first derivative is2*scale*abs(x); its one-sided difference quotients at0 are opposite nonzero values, so the second derivative there does not exist. Higher-order existence cannot be inferred from one preceding order. The exact result, retaining the requested order, is (25, f(x)=scale*x*abs(x), 0, Yes, 0, No, first_derivative_exists_at_zero_but_second_derivative_has_opposite_one_sided_limits).
Reviewed step-by-step derivation
Higher-order derivative · qf-mod42-higher-missing-next-order-boundary · candidate
Canonical question JSONL · English JSONL · Validation report