# How to compute min and max for two variable functions?

I have many functions with two variables $x$ and $y$, for which I need to compute min and max to see if they are in the range I want them to be. For example:

$$z = 0.5 - 0.25 \cos(\pi x) - 0.25 \cos(\pi y),$$ where $x\in[0,1]$ and $y\in[0,1]$.

I tried Wolfram Alpha but for some reason it does not compute the min and max.

Thanks

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$$0.5-0.25(\cos(\pi x)+\cos(\pi y))$$ $$=0.5-0.5\cos\frac{\pi(x+y)}2\cos\frac{\pi(x-y)}2$$ Now you just need the min and max of $\displaystyle\cos\frac{\pi(x+y)}2\cos\frac{\pi(x-y)}2$ – Host-website-on-iPage Jul 9 '12 at 17:56
Is "many" about $10$ or about $10\,000$? Do you need exact bounds for the range, or would a reasonable approximation do? In the latter case computing 121 values might be the fastest solution. – Christian Blatter Jul 9 '12 at 18:15

For minimum, make $\cos \pi x$, $\cos\pi y$ as big as possible. That happens at $x=y=0$.
For maximum, make $\cos \pi x$, $\cos\pi y$ as small (big negative) as possible. That happens when $x=y=1$, since $\cos \pi=-1$.