Problem

Suppose $f^2(x)$, $g^2(x)$, and $f(x)g(x)$ are all integrable functions in the interval $[0,1]$.

(a) (10 pts) Please prove the inequality

$$\left(\int_0^1 f^2(x)\,dx\right)\left(\int_0^1 g^2(x)\,dx\right)\ge \left(\int_0^1 f(x)g(x)\,dx\right)^2.$$

(Hint: $(tf(x)+g(x))^2\ge 0$ for all $x\in[0,1]$ and $t\in\mathbb{R}$.)

(b) (10 pts) Please show that

$$\int_0^1 \frac{1}{1+x^n}\,dx\ge \frac{n+1}{n+2}$$

for all $n>0$. (Hint: use (a).)