Exercise · ZW-FIRST-ORDER-NECESSARY-CONDITION-010 · ★★★ Advanced
Find all critical points of \(f(x,y)=x^3-3x+y^2\) and say which one is a candidate for a local minimum .
Choose one option.
A \((1,0)\) and \((-1,0)\); the minimum candidate is \((1,0)\) B \((1,0)\) and \((-1,0)\); the minimum candidate is \((-1,0)\) C Only \((1,0)\); it is a local minimum D Only \((-1,0)\); it is a local minimum Show answer A. \((1,0)\) and \((-1,0)\); the minimum candidate is \((1,0)\)← Back to the question
Explanation Solving \(3x^2-3=0\) and \(2y=0\) gives \(x=\pm1\), \(y=0\). The Hessian at \((1,0)\) is positive definite, while at \((-1,0)\) it is indefinite, giving a saddle point.
A first-order necessary condition states that if \(x^*\) is a local minimizer of a differentiable function \(f\), then \(\nabla f(x^*)=0\).
\[x^* \text{ mín. local} \Rightarrow \nabla f(x^*) = 0\]
More practice problems: first-order necessary condition