Analysis | Part IA, 2006

State and prove the Intermediate Value Theorem.

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

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


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

or otherwise, show that there is a subinterval [a,b][a,b]\left[a^{\prime}, b^{\prime}\right] \subseteq[a, b] such that


Deduce that there exists c(a,b)c \in(a, b) with f(c)=kf^{\prime}(c)=k. [You may assume the Mean Value Theorem.]

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