Paper 1, Section II, D

Analysis I | Part IA, 2017

(a) State the Intermediate Value Theorem.

(b) Define what it means for a function f:RRf: \mathbb{R} \rightarrow \mathbb{R} to be differentiable at a point aRa \in \mathbb{R}. If ff is differentiable everywhere on R\mathbb{R}, must ff^{\prime} be continuous everywhere? Justify your answer.

State the Mean Value Theorem.

(c) Let f:RRf: \mathbb{R} \rightarrow \mathbb{R} be differentiable everywhere. Let a,bRa, b \in \mathbb{R} with a<ba<b.

If f(a)yf(b)f^{\prime}(a) \leqslant y \leqslant f^{\prime}(b), prove that there exists c[a,b]c \in[a, b] such that f(c)=yf^{\prime}(c)=y. [Hint: consider the function gg defined by


if xax \neq a and g(a)=f(a).]\left.g(a)=f^{\prime}(a) .\right]

If additionally f(a)0f(b)f(a) \leqslant 0 \leqslant f(b), deduce that there exists d[a,b]d \in[a, b] such that f(d)+f(d)=yf^{\prime}(d)+f(d)=y.

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