Theorem: Let f be continuous on [ a , b ] and assume f ( a ) ≠ f ( b ). Then for every λ such that f ( a ) &lt; λ &lt; f ( b ), there exists a c ∈ ( a , b ) such that f ( c ) = λ.Question:Suppose that f : [ 0 , 1 ] → [ 0 , 2 ] is continuous. Use the Intermediate Value Theorem to prove that their exists c ∈ [ 0 , 1 ] such that: f ( c ) = 2 c 2 Attempt:I know that when we have the condition were f : [ a , b ] → [ a , b ], the method to prove that c exits, is the same method you would use to prove the fixed point theorem.Unfortunately I don't have an example in my notes when we have f : [ a , b ] → [ a , y ]. How would I use the IVT to answer the original question?

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Theorem: Let   f be continuous on   [  a  ,    b  ] and assume   f  (  a  )  ≠  f  (  b  ). Then for every   λ such that   f  (  a  )  &amp;lt;  λ  &amp;lt;  f  (  b  ), there exists a   c  ∈  (  a  ,    b  ) such that   f  (  c  )  =  λ.Question:Suppose that   f  :  [  0  ,  1  ]  →  [  0  ,  2  ] is continuous. Use the Intermediate Value Theorem to prove that their exists   c  ∈  [  0  ,  1  ] such that:  f  (  c  )  =  2      c    2  Attempt:I know that when we have the condition were   f  :  [  a  ,  b  ]  →  [  a  ,  b  ], the method to prove that c exits, is the same method you would use to prove the fixed point theorem.Unfortunately I don&#039;t have an example in my notes when we have   f  :  [  a  ,  b  ]  →  [  a  ,  y  ]. How would I use the IVT to answer the original question?

pokajalaq1 · Accepted Answer

Hint:Consider the function   g  (  x  )  =  f  (  x  )  −  2      x    2  .Some details:  g  (  0  )  =  f  (  0  )  ≥  0 since the range of   f is contained in [0,2]. Similarly,   g  (  1  )  =  f  (  1  )  −  2  ≤  0. Furthermore,   g is continuous since   f is. Now, either   g  (  0  ) or   g  (  1  )  ==  0, and there&#039;s nothing to prove. Or   g  (  0  )  &amp;gt;  0,   g  (  1  )  &amp;lt;  0. The Intermediate value theorem assures there exists   c  ∈  (  0  ,  1  ) such that   g  (  c  )  =  0.

Theorem: Let f be continuous on [a,b] and assume f(a)!=f(b). Then for every lambda such that f(a)<lambda<f(b), there exists a c in (a,b) such that f(c)=lambda. Question: Suppose that f:[0,1]->[0,2] is continuous. Use the Intermediate Value Theorem to prove that their exists c in [0,1] such that:

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