let f : [ 0 , a ] → R of class C 1 such that f ( 0 ) = 0 show that ∃ c ∈ ] 0 , a [ , f ′ ( c ) = 2 f ( a ) + a f ′ ( a ) 3 a First I apply mean value theorem then ∃ b ∈ ] 0 , a [ , f ′ ( b ) = f ( a ) − f ( 0 ) a − 0 = f ( a ) a then f ( a ) = a f ′ ( b )so 2 f ( a ) + a f ′ ( a ) 3 a = 2 3 f ′ ( b ) + 1 3 f ′ ( a )How to prove that 2 3 f ′ ( b ) + 1 3 f ′ ( a ) is between f ′ ( a ) and f ′ ( b )?

Question

let   f  :  [  0  ,  a  ]  →      R   of class       C    1   such that   f  (  0  )  =  0 show that  ∃  c  ∈  ]  0  ,  a  [  ,      f    ′    (  c  )  =            2      f      (      a      )      +      a              f        ′            (      a      )              3      a      First I apply mean value theorem then   ∃  b  ∈  ]  0  ,  a  [  ,      f    ′    (  b  )  =            f      (      a      )      −      f      (      0      )              a      −      0        =            f      (      a      )        a   then   f  (  a  )  =  a      f    ′    (  b  )so            2      f      (      a      )      +      a              f        ′            (      a      )              3      a        =      2    3        f    ′    (  b  )  +      1    3        f    ′    (  a  )How to prove that       2    3        f    ′    (  b  )  +      1    3        f    ′    (  a  ) is between       f    ′    (  a  ) and       f    ′    (  b  )?

Claudia Woods · Accepted Answer

2    =          2      1        ;          4    =          2      2        ;          8    =          2      3        ;          16    =          2      4        ;          ⋯          ;          2      n

akuzativo617 · Accepted Answer

If       f    ′    (  b  )  =      f    ′    (  a  ), it is easy. Otherwise, define  g  (  x  )  =      f    ′    (  x  )  −            [            2    3        f    ′    (  b  )  +      1    3        f    ′    (  a  )            ]        .Then   g  (  x  ) is continuous in   [  b  ,  a  ] and  g  (  b  )  g  (  a  )  =  −      2    9    (      f    ′    (  b  )  −      f    ′    (  a  )      )    2    &amp;lt;  0.By the IMVT, there exists   c  ∈  (  b  ,  a  ) such that   g  (  c  )  =  0 or      f    ′    (  c  )  =      2    3        f    ′    (  b  )  +      1    3        f    ′    (  a  )  .

let f:[0,a]->RR of class C^1 such that f(0)=0 show that EEc in ]0,a[,f′(c)=(2f(a)+af′(a))/(3a)

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