Question

The function f(x)= x²/³ on [-8,8] does not satisfy the conditions of the mean-value theorem because...

A. f(0) is not defined.

B. f(x) is not continuous on [-8,8].

C. f ‘ (-1) does not exist.

D. f(x) is not defined for x < 0.

E. f ‘ (0)does not exist.

Answer 1

The function f(x) = x²/³ on [-8,8] does not satisfy the conditions of the mean-value theorem because the derivative at x = 0 does not exist, creating a cusp at this point.

The correct option is B.

Step-by-step explanation:

The function f(x) = x²/³ on [-8,8] does not satisfy the conditions of the mean-value theorem because E. f ' (0) does not exist. The mean-value theorem requires the function to be continuous on the closed interval and differentiable on the open interval.

While the function is defined and continuous for all x in [-8,8], the derivative at x = 0 does not exist because the function abruptly changes direction at this point, which causes a cusp.

This makes the slope (derivative) at this point undefined, violating one of the key conditions of the mean-value theorem.

The correct option is B.