Prove that point $T(0, 0)$ is saddle point of $f(x,y) = xy$

Question

how can we prove that point $T(0, 0)$ is saddle point of a function $$f(x,y) = xy$$

I know that we can use Hessian matrix for determining minima/maxima and saddle points of functions, but I am confused with this example since second derivatives of this function are constants.

Thank you in advance

Answer 1

The Hessian matrix is $\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$. Its eigenvalues are $\pm1$. Since it has an eigenvalue which is positive and another one which is negative, the function has a saddle point.

Answer 2

Note that $f(x,x)=x^2$ which has a minimum at $(0,0)$ and $f(x,-x)=-x^2$ which has a maximum at $(0,0).$

Thus $(0,0)$ is a saddle point.

Answer 3

It depends on your definition of a saddle point. Maybe it is clearer if you write it as $$ 2xy=\frac12(x+y)^2-\frac12(x-y)^2=u^2-v^2. $$ The orthogonal change of the variables $(x,y)\mapsto (u,v)$ is a rotation by $\pi/4$.

Answer 4

$f(0,0)=0$ and $xy$ is neither negative, nor positive in any neighbourhood of $(0,0)$, but can be both.