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Math Help - Mean Value Problem

  1. #1
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    Mean Value Problem

    Left f(x) be a function that is continuous on the interval [-5,5] with continuous 1st and 2nd derivatives on (-5,5). f(-5)=3, f(0)=-2 and f(5)=8.

    1. Show that there is a point a in (-5, 0) such that f'(a) <0

    2. Show that there is a point b in (0, 5) such that f'(b) <0

    3. Using the results from parts 1 & 2, must f have a critical number in (-5, 5)? Explain why f must have a critical number or give an example of a function that satisfies the criteria but has no critical number.
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  2. #2
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    [QUOTE=ebonyscythe;83912]Left f(x) be a function that is continuous on the interval [-5,5] with continuous 1st and 2nd derivatives on (-5,5). f(-5)=3, f(0)=-2 and f(5)=8.

    1. Show that there is a point a in (-5, 0) such that f'(a) <0
    Intermediate value theorem on f' applied to -5 and 0 because f'(-5)=3 and f'(0)=-2.

    2. Show that there is a point b in (0, 5) such that f'(b) <0
    I think you mean f'(b)>0 it is the same idea.
    3. Using the results from parts 1 & 2, must f have a critical number in (-5, 5)? Explain why f must have a critical number or give an example of a function that satisfies the criteria but has no critical number.
    Intermediate value theorem again. This time on 'a' and 'b'.
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