Paper 1, Section II, D

Analysis I | Part IA, 2019

State and prove the Intermediate Value Theorem.

State the Mean Value Theorem.

Suppose that the function gg is differentiable everywhere in some open interval containing [a,b][a, b], and that g(a)<k<g(b)g^{\prime}(a)<k<g^{\prime}(b). By considering the functions hh and ff defined by

h(x)=g(x)g(a)xa(a<xb),h(a)=g(a)h(x)=\frac{g(x)-g(a)}{x-a}(a<x \leqslant b), \quad h(a)=g^{\prime}(a)


f(x)=g(b)g(x)bx(ax<b),f(b)=g(b),f(x)=\frac{g(b)-g(x)}{b-x}(a \leqslant x<b), \quad f(b)=g^{\prime}(b),

or otherwise, show that there is a subinterval [α,β][a,b][\alpha, \beta] \subseteq[a, b] such that


Deduce that there exists c(a,b)c \in(a, b) with g(c)=kg^{\prime}(c)=k.

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