I'm new to mathematical proofs, and I have just covered the Intermediate Value Theorem. I have tried to practice my understanding of the theorem, but I have encountered a question that I'm not sure how to approach.The question is:Assume that f and g are continuous on the interval [0,1] and 0 ≤ f ( x ) ≤ 1 for all x ∈ [ 0 , 1 ]. Show that if g ( 0 ) = 0 and g ( 1 ) = 1, then there exists a c ∈ [ 0 , 1 ] such that f ( c ) = g ( c ).What I have tried doing:Introducing another function like such h ( x ) = f ( x ) − g ( x ) and applying the theorem. However, I'm not sure if that is possible, or how I should go about doing it.

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I&#039;m new to mathematical proofs, and I have just covered the Intermediate Value Theorem. I have tried to practice my understanding of the theorem, but I have encountered a question that I&#039;m not sure how to approach.The question is:Assume that   f and   g are continuous on the interval [0,1] and   0  ≤  f  (  x  )  ≤  1 for all   x  ∈  [  0  ,  1  ]. Show that if   g  (  0  )  =  0 and   g  (  1  )  =  1, then there exists a   c  ∈  [  0  ,  1  ] such that   f  (  c  )  =  g  (  c  ).What I have tried doing:Introducing another function like such   h  (  x  )  =  f  (  x  )  −  g  (  x  ) and applying the theorem. However, I&#039;m not sure if that is possible, or how I should go about doing it.

Haggar72 · Accepted Answer

Introducing another function like such h(x)=f(x)−g(x) and applying the theorem.That&#039;s exactly the right thing to do!h is continuous as   f  ,  g are. And   h  (  0  )  =  f  (  0  )  −  g  (  0  )  =  f  (  0  )  −  0  =  f  (  0  ) so   0  ≤  h  (  0  )  =  f  (  0  )  ≤  1. whereas   h  (  1  )  =  f  (  1  )  −  g  (  1  )  =  f  (  1  )  −  1. Now   0  ≤  f  (  1  ) so   −  1  ≤  f  (  1  )  −  1  =  h  (  1  )  ≤  0.Now IVT on h:   h  (  0  )  ≥  0  ≥  h  (  1  ) so there   c  ∈  [  0  ,  1  ] so that   h  (  c  )  =  0. Which would mean   h  (  c  )  =  f  (  c  )  −  g  (  c  )  =  0 and so   f  (  c  )  =  g  (  c  ).

I'm new to mathematical proofs, and I have just covered the Intermediate Value Theorem. I have tried

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