Posted by **Natalie** on Tuesday, July 22, 2014 at 3:48pm.

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 -
**bobpursley**, Tuesday, July 22, 2014 at 6:17pm
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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