Let $f, g:[0,1] \rightarrow \mathbb{R}$ be continuous functions with $g(x) \geqslant 0$ for $x \in[0,1]$. Show that

$$\int_{0}^{1} f(x) g(x) d x \leqslant M \int_{0}^{1} g(x) d x$$

where $M=\sup \{|f(x)|: x \in[0,1]\}$.

Prove there exists $\alpha \in[0,1]$ such that

$$\int_{0}^{1} f(x) g(x) d x=f(\alpha) \int_{0}^{1} g(x) d x$$

[Standard results about continuous functions and their integrals may be used without proof, if clearly stated.]