Question 27902: Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT

Original question and givens

Use mode=stationary_not_sufficient, complete model={"scale":2}. Give (mode, objective_description, constraint_description, point, stationarity_statement, stationary, actual_point_status, minimum_claim, boundary).

• G001 — Use mode=stationary_not_sufficient, complete model={"scale":2}. Give (mode, objective_description, constraint_description, point, stationarity_statement, stationary, actual_point_status, minimum_claim, 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-lagrange-multiplier-condition

Step-by-step derivation

source roles = [lagrange-boundary]

derived = [[mode, objective_description, constraint_description, point, stationarity_statement, stationary, actual_point_status, minimum_claim, boundary]]

source roles = [2]

derived = [stationary_not_sufficient, f(x,y)=-2*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum]

Reverse checks

• S002 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S001 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.

Result

(stationary_not_sufficient, f(x,y)=-2*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)

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.