How would I go about showing this question:Suppose that the function f : [ 0 , 1 ] → [ 0 , 1 ] is continuous. Use the Intermediate Value Theorem to prove that there exists c ∈ [ 0 , 1 ] such that f ( c ) = c ( 2 − c 2 )My thoughts are we can define a function say g ( x ) := f ( x ) − x ( 2 − x 2 ) and then g ( 0 ) = f ( 0 ) and then g ( 1 ) = f ( 1 ) − 1 and then I'm stuck on how to complete it to get to the point where g ( 0 ) &lt; 0 &lt; g ( 1 )so there would exist a c s.t. g ( c ) = 0 [by IVT] and thus f ( c ) = c ( 2 − c 2 )If anyone could help, would be much appreciated thank you!

Question

How would I go about showing this question:Suppose that the function   f  :  [  0  ,  1  ]  →  [  0  ,  1  ] is continuous. Use the Intermediate Value Theorem to prove that there exists   c  ∈  [  0  ,  1  ] such that  f  (  c  )  =  c  (  2  −      c    2    )My thoughts are we can define a function say   g  (  x  )  :=  f  (  x  )  −  x  (  2  −      x    2    ) and then   g  (  0  )  =  f  (  0  ) and then   g  (  1  )  =  f  (  1  )  −  1 and then I&#039;m stuck on how to complete it to get to the point where  g  (  0  )  &amp;lt;  0  &amp;lt;  g  (  1  )so there would exist a c s.t.   g  (  c  )  =  0 [by IVT] and thus   f  (  c  )  =  c  (  2  −      c    2    )If anyone could help, would be much appreciated thank you!

Charlize Manning · Accepted Answer

Try to prove instead that   g  (  0  )  ≥  0  ≥  g  (  1  ).

How would I go about showing this question: Suppose that the function f : [ 0 ,

Answered question

Answer & Explanation

New Questions in Calculus 2