My Math Forum Intermediate Value Theorem Problem

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 June 12th, 2015, 02:15 AM #1 Newbie   Joined: Jun 2015 From: England Posts: 1 Thanks: 0 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.
 June 12th, 2015, 05:54 PM #2 Global Moderator   Joined: May 2007 Posts: 6,807 Thanks: 717 General idea. Pick a pair of points x1 and x2 (> x1), where f(x1)=f(x2). All x's, where x1 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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