Paper 1, Section II, D

Analysis I | Part IA, 2020

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

f(x)=f(0)+f(0)x+f(0)2!x2++f(N)(0)N!xN+f(N+1)(θx)(N+1)!xN+1f(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<θ<10<\theta<1. Hence, or otherwise, show that if f(x)=0f^{\prime}(x)=0 for all xRx \in \mathbb{R} then ff is constant.

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

s(x)=c(x),c(x)=s(x),s(0)=0 and c(0)=1s^{\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(ax)+c(x)s(ax)s(x) c(a-x)+c(x) s(a-x) is independent of xx, (ii) s(x+y)=s(x)c(y)+c(x)s(y)s(x+y)=s(x) c(y)+c(x) s(y), (iii) s(x)2+c(x)2=1s(x)^{2}+c(x)^{2}=1.

Show that c(1)>0c(1)>0 and c(2)<0c(2)<0. Deduce there exists 1<k<21<k<2 such that s(2k)=c(k)=0s(2 k)=c(k)=0 and s(x+4k)=s(x)s(x+4 k)=s(x).

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