Question 65
Let $p$ and $q$ be two positive real numbers, $p \gt q$, and let $C$ be the set of the continuous real functions defined on the interval $[0, 1]$. Find
$$
\max_{f \in C} \int_{0}^{1} \left(x^{p} |f(x)|^{q} - x^{q} |f(x)|^{p}\right) dx
$$
and the functions which yield this maximum.
$$
\max_{f \in C} \int_{0}^{1} \left(x^{p} |f(x)|^{q} - x^{q} |f(x)|^{p}\right) dx
$$
and the functions which yield this maximum.