Exercise 1594Calculus › Multivariable calculus › Optimisation in several variablesL1Added 3 Nov 2026

Three-variable saddle point

#three variables#saddle#Hessian

Statement

Find and classify the critical point of $f(x,y,z)=yz+zx-xy$ .

Hint

$\nabla f=(z-y,z-x,y+x)$ ; Hessian is constant.

Show solutionHide solution

Critical point. Setting all partials to zero gives $x=y=z=0$ .

Hessian.
$H=\begin{pmatrix}0&-1&1\\-1&0&1\\1&1&0\end{pmatrix}.$

Determinant. $\det H=-2$ , $\operatorname{tr}H=0$ . Eigenvalues can't be all positive (product $-2$ ) or all negative (sum $0$ ).

Conclusion. The origin is a saddle point.