# Question 23002: A First Derivative May Exist Where the Second Does Not

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0045-23002 · q-20261014-advance-20261111-r02-higher-order-derivative-0077



## Original question and givens

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).


- G001 — 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).





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-higher-order-derivative](https://ocme.evemisslab.com/en/mko/mko-higher-order-derivative/)





## Step-by-step derivation

S001 · impliesBefore

source operation=higher-missing-next-order-boundary;original snapshot and mathematics retained

After

explicit ordered source output fields=[scale, function_description, point, first_derivative_exists, first_derivative_value, second_derivative_exists, boundary]

Conditions: none

Retain the historical interface marker while listing source fields without JavaScript.Tags and No retain their claim scope;source-protocol ordering is not a mathematical proof.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

ORIGINAL nonzero scale=2;WHOLE f(x)=scale*x*abs(x),f(0)=0

After

for h!=0,(f(h)-f(0))/h=scale*abs(h) tends0;first derivative at0 exists and equals0

Conditions: none

Use the difference quotient for all h approaching0,not finite samples;both signs are bounded by the same vanishing absolute value.

Step identifiers and dependenciesRule: higher_first_derivative_at_zero_from_difference_quotient · Uses: G001, S001





S003 · impliesBefore

ORIGINAL f(x)=(2)x*abs(x);for x!=0,f_prime(x)=2*(2)*abs(x);f_prime(0)=0

After

next difference quotient at0 has right limit=4,left limit=-4;nonzero scale makes them unequal,so second derivative at0 does NOT exist

Conditions: none

After establishing the first derivative,check its two one-sided difference quotients.First-order existence guarantees neither the next derivative nor analyticity or Taylor-series convergence.

Step identifiers and dependenciesRule: higher_second_derivative_opposite_one_sided_limits · Uses: G001, S002









## Reverse checks


- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(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 full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 4a0d1f15aafa1ef2e7333ba8ccb30f3d16d01f6a33cc5625a39fe43de69a4537
- candidate record SHA-256: c7d2b5d434ebba544cd658088f0415d0be87d119e24d1b51f30f875215b195e1
- semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438
- question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261111-r02-higher-order-derivative-0077.json)