MathDB

6

Part of 2019 IMC

Problems(1)

Two continuous functions

Source: IMC 2019 Day 2 P6

7/31/2019
Let f,g:R→Rf,g:\mathbb R\to\mathbb R be continuous functions such that gg is differentiable. Assume that (f(0)−g′(0))(g′(1)−f(1))>0(f(0)-g'(0))(g'(1)-f(1))>0. Show that there exists a point c∈(0,1)c\in (0,1) such that f(c)=g′(c)f(c)=g'(c).
Proposed by Fereshteh Malek, K. N. Toosi University of Technology
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