Question about Hessian matrix and its application to find maximum

Question

If I have a function $f(x.y)=x^2+xy$

And I want to find out maximum, minimum or saddle points.

If I find out Hessian matrix, entries of this matrix is constant.

Then, it seems like that we can not use hessian matrix test to find out extremum points.?

Is that right?

even though stationary points exists.

Answer 1

The hessian of this function will be $$H=\begin{bmatrix} 1 & 1/2\\ 1/2 & 0 \end{bmatrix}$$Clearly the matrix is not positive definite (as corrected by @dawg in the comment) the eigenvalues being $1/2(1\pm\sqrt{2})$. So the stationary point $(0,0)$ that you get from the first order condition, is a saddle point.

Answer 2

This function has one stationary point: $(x,y)=0$. The Hessian is given by $$ \nabla^2 f = \pmatrix{ 2& 1\\1&0}. $$ Its eigenvalues are $1\pm \sqrt2$, thus the Hessian is indefinit. Hence the stationary point is a saddle point.