Originally Posted by

**alexmahone** $\displaystyle f(x,y)=\frac{1}{4}x^4+x^2y+y^2$

Find the minimum point of $\displaystyle f$ using the Hessian matrix $\displaystyle H$.

My working:

$\displaystyle f_x=x^3+2xy=0 \implies x(x^2+2y)=0$

$\displaystyle f_y=x^2+2y=0$

The critical points of $\displaystyle f$ lie on the curve $\displaystyle x^2+2y=0$.

$\displaystyle f_{xx}=3x^2+2y$

$\displaystyle f_{yy}=2$

$\displaystyle f_{xy}=f_{yx}=2x$

$\displaystyle H=\left[ \begin{array}{cc} 3x^2+2y & 2x \\ 2x & 2\end{array} \right]$

At the critical points,

$\displaystyle H=\left[ \begin{array}{cc} 2x^2 & 2x \\ 2x & 2\end{array} \right]$

How do I proceed to show that $\displaystyle H$ is positive definite?