Find all critical points of the function $f(x,y)=\cos(x)\cos(y)$ and determine which are min, max and saddle points.

Question

I'm quite confused how to determine which are min, max and saddle points because if I do the partial derivatives I I get pairs of $\pi/2+k\pi$ and $k\pi$, which means that I have an infinite number of points.

How do I find the pattern? I am not sure how to approach it.

Answer 1

The critical points are those where both partial derivatives vanish: $$ \frac{\partial f}{\partial x}=-\sin x\cos y \qquad \frac{\partial f}{\partial y}=-\cos x\sin y $$ There are indeed infinitely many points where both derivatives vanish:

1. $\sin x=0$ and $\sin y=0$, or
2. $\cos y=0$ and $\cos x=0$,

using the fact that $\cos\alpha\ne0$ if $\sin\alpha\ne0$ and conversely.

The first solution set corresponds to $x=h\pi$ and $y=k\pi$, the second solution set corresponds to $x=\pi/2+l\pi$ and $y=\pi/2+m\pi$.

Note that $h,k,l,m$ can be any integer.

Now we can look at the Hessian: \begin{align} \frac{\partial^2f}{\partial x^2}&=-\cos x\cos y\\ \frac{\partial^2f}{\partial x\,\partial y}&=-\sin x\sin y\\ \frac{\partial^2f}{\partial y^2}&=-\cos x\cos y \end{align} so we get $$ H(x,y)=\det\begin{bmatrix} -\cos x\cos y & -\sin x\sin y \\ -\sin x\sin y & -\cos x\cos y \end{bmatrix} = \cos^2x\cos^2y-\sin^2x\sin^2y $$ What can you say about $H(h\pi,k\pi)$ and about $H(\pi/2+l\pi,\pi/2+m\pi)$?