Exercise 1593Calculus › Multivariable calculus › Optimisation in several variablesM1Added 3 Nov 2026

Classifying a 3D critical point

#Hessian#3D#saddle

Statement

Classify the critical point $(0,0,0)$ of $f(x,y,z)=x^2-y^2+z^2$ .

Hint

Hessian is diagonal with entries $(2,-2,2)$ .

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Hessian. $\operatorname{diag}(2,-2,2)$ .

Eigenvalues. $2,-2,2$ : mixed signs.

Conclusion. Saddle point at the origin.