Exercise 435Calculus › Multivariable calculus › Optimisation in several variablesL1Added 3 Nov 2026

Critical points of a polynomial

#critical points#polynomial#Hessian

Statement

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

Hint

Set $\nabla f=0$ ; classify via the Hessian.

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Gradient. $\nabla f=(2x-2,\,2y-4)$ , so the unique critical point is $(1,2)$ .

Hessian. $H=\begin{pmatrix}2&0\\0&2\end{pmatrix}$ : positive definite.

Classification. Global minimum, $f(1,2)=-5$ .

Conclusion. Minimum at $(1,2)$ with value $-5$ .