Exercise 1085Calculus › Multivariable calculus › Partial derivativesL1Added 3 Nov 2026

Finding critical points

#critical points#stationary#gradient

Statement

Find the critical points of $f(x,y)=x^2+y^2-2x-4y$ .

Hint

Set both partial derivatives to zero.

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Gradient. $\nabla f=(2x-2,2y-4)$ .

Critical point. $2x-2=0\Rightarrow x=1$ ; $2y-4=0\Rightarrow y=2$ . So $(1,2)$ .

Nature. Rewrite $f=(x-1)^2+(y-2)^2-5$ : minimum at $(1,2)$ .

Conclusion. The unique critical point is $(1,2)$ , a global minimum.