Mathematical Analysis integral inequality

Question

Let $f$ be a nonnegative continuous function on $[0,1]$ , and $f$ nondecrease. Then for any $0<\alpha<\beta<1$ , we have $\int_{0}^{1}f(x)dx\geq\frac{1-\alpha}{\beta-\alpha}\int_{\alpha}^{\beta}f(x)dx$ , and $\frac{1-\alpha}{\beta-\alpha}$ is the biggest number satisfying the inequality.

Answer 1

Hint:

Show that the average value of $f(x)$ on $[\alpha,1]$ is at least the average value of $f(x)$ on $[\alpha,\beta]$.