TITLE: Find maximum value of $\int_{0}^{1}\left(f(x)\right)^3dx$
QUESTION [6 upvotes]: Find maximum possible value of $\int_{0}^{1}\left(f(x)\right)^3dx$ given that $-1\leq f(x)\leq 1$ and that $\int_{0}^{1}f(x)dx=0$.
My attempt:
I tried to guess such functions which could satisfy all the above conditions but could not arrive at any conclusion.

REPLY [2 votes]: A somewhat more conceptual approach is the following.  
The set of points 
$S=\{(\int_0^1f(x)\,dx,\int_0^1(f(x))^3\,dx): |f(x)|\le 1 \text{ on } [0,1]\}\subset\mathbb R^2$, is exactly the set of points representable as $(E[X],E[X^3])$ for a random variable $X$ for which $P(|X|\le1)=1$.  This, in turn, is the convex hull of the  set $C=\{(x,x^3):|x|\le 1\}$, and the desired answer is the $y$-coordinate of the intersection of the $y$ axis with the upper envelope of $S$.  
A sketch should make it obvious that the upper envelope of $S$ is the union of a piece of $C$ stretching from $(-1,-1)$ to $p$, and of the line segment connecting $p$ to $(1,1)$, where $p$ is the unique point of tangency to $C$ of a line passing through $(1,1)$.  Simple calculus verifies that $p=(-1/2,-1/8)$.  The line segment from $p$ to $(1,1)$ intersects the $y$ axis at $(0,1/4)$, from which the answer $1/4$ is read off.