# 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.

## Answers and Replies

Dick
Science Advisor
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
Science Advisor
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.