Final answer:
The function f(x)=x^(2/3) does not fulfill the conditions of the mean value theorem on [-8,8] because it is not differentiable at x=0.
Step-by-step explanation:
The mean value theorem states that for any function that is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), there exists at least one c in the interval (a,b) where:
f'(c) = (f(b) - f(a)) / (b - a).
The function f(x)=x(2/3) does not satisfy the mean value theorem on the interval [-8,8] because it is not differentiable at x=0. The derivative of x(2/3) involves a term with x in the denominator, becoming undefined at x=0. Thus, there might not exist a number c in the interval where the mean value theorem holds.