Exercise 1796Calculus › Multivariable calculus › Optimisation in several variablesL1Added 3 Nov 2026

The Hessian matrix

#Hessian#second derivatives#quadratic form

Statement

Compute the Hessian of $f(x,y)=x^2+4xy+3y^2$ at $(0,0)$ and classify the critical point.

Hint

The Hessian is the matrix of second partials; classify via its eigenvalues.

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Partial derivatives. $f_x=2x+4y$ , $f_y=4x+6y$ ; $f_{xx}=2$ , $f_{yy}=6$ , $f_{xy}=4$ .

Hessian. $H=\begin{pmatrix}2&4\\4&6\end{pmatrix}$ .

Eigenvalues. $\det(H-\lambda I)=(2-\lambda)(6-\lambda)-16=\lambda^2-8\lambda-4$ ; roots $\lambda=4\pm2\sqrt5$ . One positive, one negative.

Conclusion. The origin is a saddle point (Hessian is indefinite).