# If $f$ is continuous and $3\geq f(x)\geq 1$ for all $x\in[0,1]$, show the integral inequality.

If $$f$$ is continuous and $$3\geq f(x)\geq 1$$ for all $$x\in [0,1]$$, show that $$1 \leq \int_0^1f(x)dx\int_0^1\Bigg(\frac{1}{f(x)}\Bigg)dx \leq \frac{4}{3}.$$

Thanks!

By C-S $$\int\limits_0^1f(x)dx\int\limits_0^1\frac{1}{f(x)}dx \geq \left(\int\limits_0^11dx\right)^2=1.$$ For the proof of the right inequality use the Schweitzer's inequality.