State and prove the Mean Value Theorem.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that, for every $x \in \mathbb{R}, f^{\prime \prime}(x)$ exists and is non-negative.

(i) Show that if $x \leqslant y$ then $f^{\prime}(x) \leqslant f^{\prime}(y)$.

(ii) Let $\lambda \in(0,1)$ and $a<b$. Show that there exist $x$ and $y$ such that

$$f(\lambda a+(1-\lambda) b)=f(a)+(1-\lambda)(b-a) f^{\prime}(x)=f(b)-\lambda(b-a) f^{\prime}(y)$$

and that

$$f(\lambda a+(1-\lambda) b) \leqslant \lambda f(a)+(1-\lambda) f(b) .$$