Exercise · ZW-FIRST-ORDER-NECESSARY-CONDITION-004 · ★★☆ Intermediate
If \(\nabla f(x^*)\neq 0\), can \(x^*\) be a local minimum of \(f\)?
Choose one option.
Show answer
- C. No, it is impossible in any case
Explanation
If the gradient is nonzero, there is a descent direction, so the point cannot be a local minimum.
Review the concept: First-order necessary condition
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
- ZW-FIRST-ORDER-NECESSARY-CONDITION-001 What must the gradient of a differentiable function satisfy at an interior local minimum?
- ZW-FIRST-ORDER-NECESSARY-CONDITION-002 State the first-order necessary condition for a local minimum of f at x^*.
- ZW-FIRST-ORDER-NECESSARY-CONDITION-003 What is the name of the condition requiring the gradient to vanish at a local minimum?
- ZW-FIRST-ORDER-NECESSARY-CONDITION-005 What kind of candidate points does the first-order condition identify for an unconstrained differen…
- ZW-FIRST-ORDER-NECESSARY-CONDITION-006 Solve ∇ f(x,y)=0 for f(x,y)=(x-2)^2+(y+1)^2.
- ZW-FIRST-ORDER-NECESSARY-CONDITION-007 Find the critical point of f(x,y)=x^2-4x+y^2+6y.
- ZW-FIRST-ORDER-NECESSARY-CONDITION-008 For f(x,y)=x^2+xy+y^2-3x, what is ∂ f/∂ x evaluated at (1,1)?
- ZW-FIRST-ORDER-NECESSARY-CONDITION-009 For f(x,y)=x^2+xy+y^2-3x, at which point does the gradient vanish?
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