# Basics: find maxima

I'm completely new to Mathematica and recently tried to maximize the function x^(1-a)*(v-x)^a with respect to x under the constraints 1>a>0 and v>=x>=0.

As far as I know the maximum should be at x=v-av.

Maximize[{x^(1 - a) (v - x)^a , 1 > a > 0 && v >= x >= 0}, x]


I'd be grateful if someone can explain me why, and how the command should look like if I want to get a result with one command.

• You could try to find the zeros of the first derivative ... Aug 3 '16 at 15:24
• I know that I can do: D[x^(1-a)*(v-x)^a], x] and then Solve[-a (v - x)^(-1 + a) x^(1 - a) + (1 - a) (v - x)^a x^-a == 0 && 1 >= a >= 0 && v >= x >= 0, x, Reals] But I'd like to know why "Maximize" does not work. I feel like this is fundamental. Aug 3 '16 at 15:28
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• Have you asked Wolfram about it? As you say, it seems fundamental. Aug 3 '16 at 15:52

Maximize[] and like functions often have problems with variable or non-integer exponents.

Even simple functions of the like fail.

But there's nothing wrong with remembering our high-school algebra. There are always going to be cases where we need human interaction (luckily) This function only has a maximum

Manipulate[Plot[x^(1 - a) (v - x)^a, {x, 0, v}], {a, 0, 1}, {v, 1, 4}]


Remember, the maximum can be found when the derivative is zero:

Reduce[D[x^(1 - a) (v - x)^a, x] == 0]


a v (v - a v)^a != 0 && x == v - a v

In other words, $x=v-a v$

Manipulate[
Show[Plot[x^(1 - a) (v - x)^a, {x, 0, v}],
Graphics[{Red, Dashed, Line[{{v - a v, 0}, {v - a v, 2}}]}]], {a, 0,
1}, {v, 1, 4}]


• Thank you Feyre! Unfortunately I have a follow-up question :) In the example I chose a simple function. How should I proceed if it gets a bit more complicated? For example: $$\max_{x_1} x_1^{(1-\alpha)}*x_2^{\alpha/2}*x_3^{\alpha/2} \: \textrm{s.t.} \: v=x_1+x_2+x_3; \alpha \in [0;1]; x_1,x_2,x_3\in [0;v]$$ I think (please correct me if I'm wrong) this should have the same maximum as before because at maximum $x_2=x_3$. But assume we don't know that. With Reduce[D[x1^(1 - a)*(v - x1 - x3)^(a/2)*(v - x1 - x2)^(a/2), x1] == 0, x1] Mathematica does not give me $x = v - a v$ as solution. Aug 3 '16 at 17:23
• And what if the optimization problem gets even more complicated? Is there no way to maximize if the exponents are non-integer? Aug 3 '16 at 17:35
• @Manuel Even very simple functions can't be maximized then. If you can't use numeric functions, unfortunately I think you just need to work these out. Try Reduce[-(1/2) a x1^(1 - a) (v - x1 - x2)^( a/2) (v - x1 - x3)^(-1 + a/2) - 1/2 a x1^(1 - a) (v - x1 - x2)^(-1 + a/2) (v - x1 - x3)^( a/2) + (1 - a) x1^-a (v - x1 - x2)^(a/2) (v - x1 - x3)^(a/2) == 0] Then solve for x1 Aug 3 '16 at 17:46