How can I find the minimum and maximum?

Question

Lets have the following equation:

$f(x,y,y) = cos(x)^2+\frac{1}{1+x^3}+y^3+z$

I would like to find the minimum and maximum where

$-2<x<2$

$-1<y<1$

$-2<z<1$

How can I do that, I would go with second order Taylor series and linearize around $(0,0,0)$, Than I can use the quadratic formulation.

I would solve the inequality via Matlab by using the quadprog function, but here come an other question, what is behind the quadprog function?

Answer 1

You can easily split $f(x,y,z)$ into three separate functions and then find their separate minimum/maximums. i.e. $$ g(x) = \cos(x)^2 + \frac{1}{1+x^3} \\ g(y) = y^3 \\ g(z) = z \\ f(x,y,z) = g(x)+g(y)+g(z)$$

Should be able to eyeball the rest from the graphs drawn within appropriate domains.