Let f(x,y)=√x^2+y^2. Which statement is true?

(0,0) is a critical point because ∇ f does not exist there

f has no critical points because ∇ f≠mathbf 0 always

(0,0) is not critical because f is not differentiable there

(0,0) is critical because ∇ f(0,0)=mathbf 0

Exercise · ZW-CRITICAL-POINT-MULTIVARIABLE-009 · ★★★ Advanced

Let \(f(x,y)=\sqrt{x^2+y^2}\). Which statement is true?

Choose one option.

Show answer

B. \((0,0)\) is a critical point because \(\nabla f\) does not exist there

Explanation

Away from the origin \(\nabla f\) has length 1, and at the origin it does not exist (the cone has a vertex). Since \(\nabla f\) does not exist, the origin is critical by definition.

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Review the concept: Critical point of a multivariable function

A critical point of a function \(f\) of two variables is a point \((a,b)\) of its domain where the gradient \(\nabla f\) is the zero vector or does not exist.

\[\nabla f(a,b)=\mathbf 0\quad\text{or}\quad\nabla f(a,b)\text{ does not exist}\]

Let \(f(x,y)=\sqrt{x^2+y^2}\). Which statement is true?

Explanation

Review the concept: Critical point of a multivariable function

More practice problems: critical point of a multivariable function