Functional optimization problem

[phreak]

Homework Statement

Maximize the functional $$\int_{-1}^1 x^3 g(x)$$, where g is subject to the following conditions:

$$\int^1_{-1} g(x)dx = \int^1_{-1} x g(x)dx = \int^1_{-1} x^2 g(x)dx = 0$$ and $$\int^1_{-1} |g(x)|^2 dx = 1$$.

Homework Equations

In the previous part of the problem, I computed $$\min_{a,b,c} \int^1_{-1} |x^3 - a - bx - cx^2|^2 dx$$. I'm not sure how this is related, or if it is at all.

The Attempt at a Solution

Thus far, I have only tried to look for patterns. In particular, I've tried simply looking for functions g satisfying the conditions, without trying to maximize. I've found a few, and they seem to be closely related to the exponential function. I will continue to look, but I think I may need a boost to get started. I'll be very grateful for any hints anyone can give me.

EDIT: Hours of trying to solve this, then finally posting it to PF, then trying the Cauchy-Schwartz inequality with a bit of tricky algebra and finding the solution is really frustrating.

 

[lanedance]

hi phreak, so i think this is a calculus of variations problem, have you tried using the Euler Lagrange equation with Lagrange multipliers?

 

[lanedance]

to expand on that, when you have the problem of optimising an integral for an unknown function g, where the integrand is given by f
$$\int dx f(g, g', x)$$

subject to constraints h_i
$$\int dx h_i(g, g', x) = c_i$$

then write the total equation as
$$L(g, g', x) = f(g, g', x) + \lambda_i h_i(g, g', x)$$
where the lambda's are as yet undetermined lagrange multipliers

then the the optimising function must satisfy the Euler-Lagrange equation
$$\frac{\partial L(g, g', x)}{\partial g} - \frac{d}{dx} \frac{\partial L(g, g', x)}{\partial g'} = 0$$

 

[lanedance]

note in your case a lot of thing simplify as there is no g' term in your equations