Maximum value of a function when a function of several variables, and each variable is not dependent on each other

Question

A function $f$ is of several variables. Each variable is not a function of any other variables. I'd like to find out whether global maximum point exists, and if it does, what would the values of variables look like. How do I do this?

Answer 1

If your domain is compact:

1. For points inside your domain look for points where all partial derivatives are zero.

2. For points on the boundary of your domain use Lagrange multipliers: http://en.wikipedia.org/wiki/Lagrange_multiplier

3. Pick the maximal value from the above candidates.

If your domain is not bounded then also compute limits.

Answer 2

If your function is differentiable and simple, you can find partial derivatives and solve the equations that result from setting these to zero. In other words, you can proceed as in Michael's answer.

But in practice (i.e. except for homework problems) this approach will probably give you an ugly set of non-linear equations that you won't be able to solve, except by using numerical methods. In that case, you're actually better off using numerical methods to solve the original problem. In other words, you need a numerical optimization algorithm. The best algorithm for your situation depends strongly on the specifics of your problem (whether your function is convex, or differentiable, the shape of its domain, and so on). A good place to start is here.