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f'(1)>1 implies f has a fixed point in (0,1)

Source: ISI(BS) 2010 #4

May 17, 2012
functionlimitreal analysisreal analysis unsolved

Problem Statement

A real valued function ff is defined on the interval (−1,2)(-1,2). A point x0x_0 is said to be a fixed point of ff if f(x0)=x0f(x_0)=x_0. Suppose that ff is a differentiable function such that f(0)>0f(0)>0 and f(1)=1f(1)=1. Show that if f′(1)>1f'(1)>1, then ff has a fixed point in the interval (0,1)(0,1).
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