Monday, November 21st

Problem
Let $f(x), g(x)$ be two continuous real functions such that $\int_0^1f^2(x)dx=\int_0^1 g^2(x)dx=1$. Prove that there is a real number $c$ such that $f(c)+g(c)\le 2$.

Solution
Suppose that $f(c)+g(c) > 2$ for all $c\in[0,1]$, then $\int_0^1 f(x)g(x)dx>1$, but by Cauchy-Schwarz $\int_0^1 f(x)g(x)dx\le1$ which gives a contradiction, hence such $c$ exists.