Homework Help: Functional optimization problem

1. Oct 27, 2009

phreak

1. The problem statement, all variables and given/known data

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

2. Relevant 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.

3. 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.

Last edited: Oct 27, 2009
2. Oct 28, 2009

lanedance

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

3. Oct 28, 2009

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
$$\frac{\partial L(g, g', x)}{\partial g} - \frac{d}{dx} \frac{\partial L(g, g', x)}{\partial g'} = 0$$

4. Oct 28, 2009

lanedance

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