Let f be a continuous function on [ 0 , 1 ] and differentiable on ( 0 , 1 ) with f ( 1 ) = f ( 0 ) = 0 and f ( 1 2 )=1 .Show that(1) there exist some a ∈ ( 1 2 , 1 ) such that f ( a ) = awhile this just can be proved by using the intermediate value theorem and I have solved it. But the second part made me confuse,(2) for any real number λ , there exist some point ε ∈ ( 0 , a ) such that f ′ ( ε ) − 1 = λ ( f ( ε ) − ε ) )For the second part, why there's a f ′ ( ε ) there? and how I can approach the proof of second part?

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Let   f be a continuous function on   [  0  ,  1  ] and differentiable on   (  0  ,  1  ) with   f  (  1  )  =  f  (  0  )  =  0 and   f  (      1    2    )=1 .Show that(1) there exist some a   ∈  (      1    2    ,  1  ) such that   f  (  a  )  =  awhile this just can be proved by using the intermediate value theorem and I have solved it. But the second part made me confuse,(2) for any real number   λ , there exist some point   ε  ∈  (  0  ,  a  ) such that       f    ′    (  ε  )  −  1  =  λ  (  f  (  ε  )  −  ε  )  )For the second part, why there&#039;s a       f    ′    (  ε  ) there? and how I can approach the proof of second part?

Erzrivalef6 · Accepted Answer

Can I use Mean Value Theorem (MVT) to prove part(2)??      f    ′    (  ε  )  =            f      (      ε      )      −      f      (      0      )              ε      −      0       such that we have:      f    ′    (  ε  )  =            f      (      ε      )        ε  Each side minus one at the same time      f    ′    (  ε  )  −  1  =            f      (      ε      )        ε    −  1then we have what we want:      f    ′    (  ε  )  −  1  =  λ  (  f  (  ε  )  −  ε  )

Let f be a continuous function on [ 0 , 1 ] and differentiable on ( 0

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