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).
G001 — 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).
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.
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.
ORIGINAL f(x)=(19)x*abs(x);for x!=0,f_prime(x)=2*(19)*abs(x);f_prime(0)=0
After
next difference quotient at0 has right limit=38,left limit=-38;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.
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
(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 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.