Exercise · ZW-FIRST-ORDER-NECESSARY-CONDITION-009 · ★★☆ Intermediate
For \(f(x,y)=x^2+xy+y^2-3x\), at which point does the gradient vanish?
Choose one option.
Show answer
- B. \((2,-1)\)
Explanation
From \(2x+y-3=0\) and \(x+2y=0\) we get \(x=2\), \(y=-1\).
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-004 If ∇ f(x^*)≠ 0, can x^* be a local minimum of f?
- 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-008 ZW-FIRST-ORDER-NECESSARY-CONDITION-010 →