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Math Help - continuous bounded functions

  1. #1
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    continuous bounded functions

    Let f:[a,b]---->R be Riemann integrable. Define F:[a,b]---->R
    by

    F(x)= \int_a^x f(t) dt
    iii) Prove that there exists c in[a,b] such that F(c)=sup{F(x): x in [a,b]}
    By boundedness principle a continuous functions attains a maximum and minimum so f is bounded.

    iv) Now suppose that f is continuous and that the point c from (iii) satisfies c in (a,b) what can you conclude about f(c)?

    I know from lecture note, If f is continuous at c in[a, b] then F is differentiable at c and F'(c) = f(c).
    from iii) f(c)=maximum of function so the derivative at maximum value is f'(c)=0 I have been asked for f(c) so can you help please? thanks
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  2. #2
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    Quote Originally Posted by charikaar View Post
    Let f:[a,b]---->R be Riemann integrable. Define F:[a,b]---->R
    by

    F(x)= \int_a^x f(t) dt
    iii) Prove that there exists c in[a,b] such that F(c)=sup{F(x): x in [a,b]}
    By boundedness principle a continuous functions attains a maximum and minimum so f is bounded.

    iv) Now suppose that f is continuous and that the point c from (iii) satisfies c in (a,b) what can you conclude about f(c)?

    I know from lecture note, If f is continuous at c in[a, b] then F is differentiable at c and F'(c) = f(c).
    from iii) f(c)=maximum of function so the derivative at maximum value is f'(c)=0 I have been asked for f(c) so can you help please? thanks
    Well I believe that F is a continuous function since:

    l \int_a^x f(t) dt - \int_a^c f(t) dtl <= l \int_c^x f(t) dtl <= \int_a^x lf(t)l dt<=lx-cl*sup{f(x): x in [a,b]}

    but f is bounded right? So sup{f(x): x in [a,b]} = K. Thus for all e>0, set delta = e/K

    Thus, iii) follows from the extreme value theorem.

    iv) Since c is in (a,b) we have F'(c) = 0

    But F'(x) = f(x)

    Thus, F'(c) = f(c) = 0

    Note: Done while tired and in haste
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