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Waylon Mcbride

Waylon Mcbride

Answered question

2022-05-09

Let a , b R , a < b and let f be a differentiable real-valued function on an open subset of R that contains [a,b]. Show that if γ is any real number between f ( a ) and f ( b ) then there exists a number c ( a , b ) such that γ = f ( c ).

Hint: Combine mean value theorem with the intermediate value theorem for the function ( f ( x 1 ) f ( x 2 ) ) x 1 x 2 on the set { ( x 1 , x 2 ) E 2 : a x 1 < x 2 b }.

I am having a lot of trouble trying to start on this problem.

Answer & Explanation

Lankenp19hh

Lankenp19hh

Beginner2022-05-10Added 11 answers

Let
g ( x 1 , x 2 ) = f ( x 1 ) f ( x 2 ) x 1 x 2
Then
lim x 2 a g ( a , x 2 ) = f ( a )  and  lim x 1 b g ( x 1 , b ) = f ( b )
So if ε is sufficiently small, what can you say about g ( a , a + ε ) , γ and g ( b ε , b )?

Now if you define h ( t ) = g ( a + ( b ϵ a ) t , a + ϵ + ( b a ϵ ) t ), what does the Intermediate Value Theorem tell you?

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