Non-negative Lebesgue Measurable Functions/Determining Measure of a Particular Set

Question

I'm having some difficulties trying to figure out where to even start with this problem:

Let $f$, $g$ be non-negative, measurable functions on $\left[ 0,1 \right]$ such that

$\int_0^1 f(x)dx=2$, $\;\;\;$ $\int_0^1 g(x)dx=1$, $\;\;\;$ and $\int_0^1 (f(x))^2 dx=5$.

Let $E=\{ x \in [0,1]: f(x) \geq g(x)\}$. Show that $m(E) \geq \frac{1}{5}$ ($m$ is the Lebesgue measure).

Answer 1

Hint: find a lower bound on $\int_E f(x)\; dx$, then use Cauchy-Schwarz.