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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<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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intermediate, problem, theorem 
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