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calculus

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verify that the function satisfies the hypothesis of the mean value theorem on the given interval. then find all numbers c that satisfy the conclusion of the mean value theorem.

f(x) = x/(x+2) , [1,4]

  • calculus -

    f(b)-f(a)/b-a = f'(c) MVT

    to find f(b) and f(a), just plug endpoints into original function
    f(b) = f(4) = (2/3)
    f(a) = f(1) = (1/3)

    (2/3)-(1/3)
    ----------- = f'(c)
    4 - 1

    (1/9) = f'(c)

    next, find derivative of f(x)
    f'(c) = f'(x)
    product rule
    (1/9) = (x)(x+2)^-1
    (1/9) = (x+2)^-1 - x(x+2)^-2
    (1/9) = (1/x+2) - (x/(x+2))^2
    (1/9) = (1/x+2) * (1 - (x/(x+2))

    (1/x+2) = (1/9) mult. ea s. by 9
    (9/x+2) = 1
    9 = x + 2
    7 = x

    I'm sure you can solve the other x

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