How to maximize a multivariable equation?

[DyslexicHobo]

I have an function with 4 variables. Each of the 5 variables are bounded between to real numbers. Is there an easy way to find a maximum value of the function?

Here is the equation: http://yfrog.com/jleqnp

I have MathCAD and Mathematica software available to me. I'm not very familliar with how to use Mathematica, though.

Thanks for any help.

Edit: I just realized that the function shown is actually the quotient of two functions. The left side is what I need to maximize (I(x1,x2,x3,x4) / A(x1,x2,x3,x4)). It can be treated as one function. I just didn't define it as so.

 

[hotvette]

Finding critical points (min/max/inflection) of multivariable functions is an extension of single variable functions. For single variable functions, critical points are where the derivative of the function with respect to the variable is equal to zero.

For multivariable functions, critical points are where the partial derivatives of the function with respect all independent variable is equal to zero. You thus end up with a system of equations to solve.

As a simple example, think of a bowl. Pick an arbitrary point on the bowl and imagine two planar slices a right angles through the bowl. The profile of each slice will look sort of like a parabola but the location of the minimum point in each slice will be different. Only when the slices go through the bottom of the bowl will the minimum points of the two slices be coincident.