Exercise 863Calculus › Multivariable calculus › Optimisation in several variablesL1Added 3 Nov 2026

A unique critical point that is a saddle

#saddle#unique critical#Hessian

Statement

Find and classify the critical point of $f(x,y)=xy+x^3y^2$ .

Hint

$f_x=y(1+3x^2y)$ , $f_y=x(1+2x^2y)$ .

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System. $y(1+3x^2y)=0$ and $x(1+2x^2y)=0$ . Only solution: $(0,0)$ .

Hessian at $(0,0)$ . $f_{xx}=6xy^2$ , $f_{xy}=1+6x^2y$ , $f_{yy}=2x^3$ ; $H(0,0)=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ .

Determinant. $\det H=-1<0$ : saddle point.

Conclusion. The only critical point $(0,0)$ is a saddle.