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SOLUTION 3: We are given the function f(x)=x2−x2/3 and the interval [−1,8]. This function is continuous on the closed interval [−1,8] since it is the sum of continuous functions y=x2 (polynomial) and y=x2/3=(x2)1/3 (the functional composition of continuous functions y=x2 and y=x1/3). The derivative of f is
f′(x)=2x−(2/3)x−1/3=2x−23x1/3
We can now see that f is NOT differentiable on the open interval (−1,8) since f′ is not defined at x=0. The assumptions of the Mean Value Theorem have NOT been met, so the Mean Value Theorem does not apply.
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