help with improper integral claim

Question

We are finding difficulties in solving this claim:

Let's suppose that $$ \int_a^\infty f(x)^2 dx < \infty \text{ and } \int_a^\infty f''(x)^2 dx < \infty. $$ Prove that $$\int_a^\infty f'(x)^2 dx < \infty.$$

Thank everyone who can solve this claim or give any suggestion.

Answer 1

Hint: integration by parts and Cauchy-Schwarz.