# Minimizing a multivariable function in several variables

I would like to show that a certain function is negative, to help establish asymptotic stability via a Lyapunov function for a system of differential equations.

This is exactly what I need help on:

Find the minimum of $$A[\frac{B}{N}(C+D)(E+F)+\frac{GC}{N}(F+H+J+K+L)]-(MC-A)^2-\frac{MB^2}{N}[B(E+F)+G(F+H+J+K+L)]+\frac{AC}{N}[B(E+F)+G(F+H+J+K+L)]$$

with constraint:

$N>A,B,C,D,E,F,G,H,J,K,L,M> 0 \\$

So clearly, $$-(MC-A)^2-\frac{MB^2}{N}[B(E+F)+G(F+H+J+K+L)]<0$$

What about the rest of the positive terms??

Is there any software that can do this sort of thing?

Thanks.

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First of all, are you minimising with respect to all letters in latin alphabet?=) – TZakrevskiy Jul 1 '14 at 17:41
@TZakrevskiy. No. I just used the letters to replace cumbersome notation. – Anonymous Jul 1 '14 at 17:45
Still not clear what are you trying to do. Are all those variables independent? If yes, then any CAS like Mathematica should be able to show what's going on - or, at least, give an idea of an analytical proof. – TZakrevskiy Jul 1 '14 at 17:52
Yes, they are all independent. I do not know how to use Maple or Mathematica or MATLAB for this functionality, however. – Anonymous Jul 1 '14 at 18:14