Multivariable and Vector Calculus · Constrained optimization and Lagrange multipliers
Diagnose a constrained minimum missed by multiplier equations
Problem
Minimize $f(x,y)=x$ subject to the semicubical cusp \[ y^2=x^3. \] Parameterize the entire feasible set, prove that the origin is the unique global constrained minimum, and then test the Lagrange equation $\nabla f=\lambda\nabla g$ for $g(x,y)=y^2-x^3$ at that point. Explain why failure of that equation is not a contradiction to the Lagrange multiplier theorem, and identify the failed hypothesis.
Hint
The constraint forces $x\ge0$.
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