Optimisation of continuous function in [0,1]

Hi everyone!

I've been struggling for a while with the following problem. I'd be very grateful if any of you have advice or pointers.

Basically there's a set of variables whose domain is [0,1] (extremes included) x1, ..., xn.

The function to maximise looks like this

It's a sum of products where variables can appear or not and the coefficient is 1 or -1.

Ex.

x1*x2 - x1*x3*x4 + x2*x4

more in general the function looks like

C0*x1*...*xn +

C11*x2*...*xn + ... + C1n*x1*...*xn-1 +

...

CN1*x1 + ... CNN*x2 +

C

with the constants being 1, -1 or 0

My intuition is that there the maximum of this function always belongs to the corners e.g. [1,0,0,0,1] or [1,1,1,0,0] but I haven't managed to prove it yet.

Thanks for the help.