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June 12th, 2015, 03:15 AM   #1
Joined: Jun 2015
From: England

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Intermediate Value Theorem Problem

I know this problem is to do with the IVT
but I'm not sure where to start- could somebody please give me a hint- thanks

Suppose that F: R---> R is continuous at every point. Prove that the equation
F(x) = c cannot have exactly two solutions for every value of c.
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June 12th, 2015, 06:54 PM   #2
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General idea. Pick a pair of points x1 and x2 (> x1), where f(x1)=f(x2). All x's, where x1<x<x2 have f(x) either all > f(x1) or all < f(x1). Otherwise intermediate value theorem gives a third point x0 in the interval where f(x0)=f(x1). Now the problem is reduced to showing that f(x) (assume > f(x1)) has a max, which is unique.
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