(a) State Rolle's theorem. Show that if $f: \mathbb{R} \rightarrow \mathbb{R}$ is $N+1$ times differentiable and $x \in \mathbb{R}$ then

$$f(x)=f(0)+f^{\prime}(0) x+\frac{f^{\prime \prime}(0)}{2 !} x^{2}+\ldots+\frac{f^{(N)}(0)}{N !} x^{N}+\frac{f^{(N+1)}(\theta x)}{(N+1) !} x^{N+1}$$

for some $0<\theta<1$. Hence, or otherwise, show that if $f^{\prime}(x)=0$ for all $x \in \mathbb{R}$ then $f$ is constant.

(b) Let $s: \mathbb{R} \rightarrow \mathbb{R}$ and $c: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable functions such that

$$s^{\prime}(x)=c(x), \quad c^{\prime}(x)=-s(x), \quad s(0)=0 \quad \text { and } \quad c(0)=1$$

Prove that (i) $s(x) c(a-x)+c(x) s(a-x)$ is independent of $x$, (ii) $s(x+y)=s(x) c(y)+c(x) s(y)$, (iii) $s(x)^{2}+c(x)^{2}=1$.

Show that $c(1)>0$ and $c(2)<0$. Deduce there exists $1<k<2$ such that $s(2 k)=c(k)=0$ and $s(x+4 k)=s(x)$.