calculus

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Find a point c satisfying the conclusion of the Mean Value Theorem for the function f(x)= x^1/3 on the interval [1,8]

I got f'(c)= 1/7 but am not sure where to go from there.

  • calculus -

    f'(x)=1/3 (x)^-2/3

    f (1)=1^1/3=1
    f (8) = 2

    so, the point c must be such that
    f'(c)= (f(8)-f(1))/(8-1)=(2-1)/7=1/7

    but f'(c)=1/3 (c)^-2/3 and f'(c)=1/7 so
    1/7=1/3(c^-2/3)
    3/7=c^-2/3
    take each side to the 3/2 power
    c^2/3=7/3
    c= cube root (7/3)^2 = 1.76

    so, the conclusion of the theorem is borne out.

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