$\int_0^1 f(x)dx \times \int_0^1{\frac{1}{f(x)}}dx \geq 1$

Question

Assume that $f$ is a measurable function on the interval $[0,1]$ such that $0<f(x)<\infty $ for $ x \in [0,1]$ Prove that

$$\int_0^1 f(x)dx \times \int_0^1{\frac1{f(x)}}dx \geq 1$$

I am trying to see on how can I apply holder's inequality to solve this porblem , but I have haven't had a breakthrough so far. Please let me know if I am thinking on the right track or if I should try to think of some other way of proving this inequality

Answer 1

Use Cauchy Schwarz with $\sqrt{f(x)}/\sqrt{f(x)}$.