Question 112
Let $f : [0, \infty) \to \mathbb{R}$ be a differentiable function, with continuous derivative, such that $f(0) = 0$, and $0 \le f'(x) \le 1$, for any $x \gt 0$. Prove that
$$
\int_{0}^{a} f(t)^{2n+1} dt \le (n+1) \cdot \left( \int_{0}^{a} f(t)^{n} dt \right)^{2},
$$
for any $a \gt 0$ and $n \in \mathbb{N}^*$.
$$
\int_{0}^{a} f(t)^{2n+1} dt \le (n+1) \cdot \left( \int_{0}^{a} f(t)^{n} dt \right)^{2},
$$
for any $a \gt 0$ and $n \in \mathbb{N}^*$.