# Maximizing a set of functions

I am working on a paper for a class, and I've come to somewhat of a block. I'll keep the question general.

If I have three non-linear real valued functions,

(1) $$f_1(x)$$
(2) $$f_2(x,W,H)$$
(3) $$f_3(x,W,H)$$

that form a function:

$$F(x,W,H) = f_1 + f_2 + f_3$$

How would I maximize $$F(x,W,H)$$.

Lagrange multipliers are ringing a bell... but before I get too invested in an idea, I would like the proper road to travel down. So If someone could point me in the right direction that would be awesome.

Dick
Homework Helper
You're just minimizing the sum of the functions. Take partial derivatives of the sum wrt to x, W and H and set them all to zero.

Are f1,f2,f3 convex?

HallsofIvy
Homework Helper
ARE W and H variables or are they constants? I see no reason to use "Lagrange multipliers" because you have no constraints.

Are f1,f2,f3 convex?

I am not actually sure. They are mixed with a lot of different terms (sinh, cosh, ...), so it is hard (for me at least) to get an idea of what they look like.

ARE W and H variables or are they constants? I see no reason to use "Lagrange multipliers" because you have no constraints.

I've recently found out that W corresponds to some constants in the functions, so I can no longer vary it.

So I will have the following:

f(x,W) = const = f1(x)+f2(x,H)+f3(x,H)

x and W both represent the length of a device. I am trying to find lengths that that maximize f(x,W), which actually represent a current.

Thanks for the help everyone.

The expressions were too complicated to take derivatives of and solve in such a way (too time consuming at least). I ended up writing a brute force algorithm to try all possible values (from a pool of "intelligent" guesses) to maximize the function.

I appreciate the help.