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Math Help - FTC w/ Chain Rule and Concavity

  1. #1
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    FTC w/ Chain Rule and Concavity

    The graph of f(x) = [0,x] integral (15t^2 - 2t^3 + 24) dt is concave up on (a,b). Find b - a

    so would f(x) = 15x^2 - 1/2x^4 + 24x?
    and then I take the derivative of that which would be 15x^2 - 2x^3 + 24 and set it equal to zero? this is confusing . . .
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  2. #2
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    f(x) = \int_0^x 15t^2 - 2t^3 + 24 \, dt

    f'(x) = 15x^2 - 2x^3 + 24

    f''(x) = 30x - 6x^2

    if f(x) is concave up on (a,b), then f''(x) > 0 on (a,b)

    30x - 6x^2 > 0

    6x(5 - x) > 0

    f''(x) > 0 on (0,5) ... b-a = 5
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  3. #3
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    oh shoot! i forgot second derivative was to determine concavity. man winter break haha. thanks !
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