Question 96
Given a positive real number $a$, determine the minimum of the expression
$$
\left( \int_{0}^{1} f(x) \, dx \right)^{2} - (a + 1) \int_{0}^{1} x^{2a} f(x) \, dx
$$
may achieve, as $f$ runs through the class of all concave functions $f : [0, 1] \to \mathbb{R}$ such that $f(0) = 1$.
$$
\left( \int_{0}^{1} f(x) \, dx \right)^{2} - (a + 1) \int_{0}^{1} x^{2a} f(x) \, dx
$$
may achieve, as $f$ runs through the class of all concave functions $f : [0, 1] \to \mathbb{R}$ such that $f(0) = 1$.