Question

I’m using Mathematica 8 to find an analytic solution to the max of an expression. When I use the Maximize command to try to find a solution, it just repeats what I entered, implying that Mathematica doesn’t know how to do it. I’ve narrowed down the problem to this: it seems like if there is an exponent that is a parameter, Maximize doesn’t work. Here’s an example. This is the likelihood function from a Bernoulli trial, where a and b are the successes and failures.

Maximize[{t^a*(1 - t)^b, {t >= 0, t <= 1, a > 0, b > 0}}, {t}]

What I would like to get as a solution is a/(a+b) in this case. If I provide constants like 3 and 2 instead of a and b then it finds the solution.

Is there a different way to specify the expression or the constraints so that Mathematica can find a maximum to expressions with exponents that are parameters? I feel like there’s something I’m missing because this is so simple.

Answer 1

I’ve been playing with it, i.e. moving conditions, changing expression form, removing conditions, and I can’t get Maximize to behave, either. However, this can be solved directly, as follows

Solve[ D[ t^a (1 - t)^b, t ] == 0, t]

which gives, as you said, {{t -> a/(a + b)}}. Sometimes Reduce can be used to help understand why a function like Maximize misbehaves by giving a more complete picture of the solution space. It is invoked like Solve, as follows

Reduce[ D[ t^a (1 - t)^b, t ] == 0, t]

giving

((-1 + t) t != 0 && a == 0 && b == 0) || 
 (a + b != 0 && a b != 0 && t == a/(a + b)) || 
 (Re[b] > 1 && t == 1) || 
 (Re[a] > 1 && t == 0)

which isn’t all that helpful, in this case.