Determine whether Rolle's Theorem can be applied to f on the closed interval
[a, b].
(Select all that apply.)
f(x) = x^2/3 − 2, [−8, 8]
1) Yes, Rolle's Theorem can be applied.
2)No, because f is not continuous on the closed interval [a, b].
3)No, because f is not differentiable in the open interval (a, b).
4)No, because f(a) ≠ f(b).
the last three answers are all false.
To determine whether Rolle's Theorem can be applied to a function on a closed interval, we need to check the following conditions:
1) Continuity: The function f(x) must be continuous on the closed interval [a, b].
2) Differentiability: The function f(x) must be differentiable on the open interval (a, b).
3) Values at endpoints: The values of f(x) at the endpoints, f(a) and f(b), must be equal.
Let's check each condition for the given function f(x) = x^(2/3) - 2 on the closed interval [-8, 8].
1) Continuity:
The function f(x) = x^(2/3) - 2 is a polynomial function combined with a constant function, which is continuous for all real numbers, including the interval [-8, 8]. Hence, f(x) is continuous on the closed interval [a, b].
2) Differentiability:
The derivative of f(x) = x^(2/3) - 2 can be found by using the power rule of differentiation. The derivative is f'(x) = (2/3)x^(-1/3). This derivative is defined for all real numbers except x = 0. So, the function f(x) is differentiable on the open interval (-8, 8), except at x = 0.
3) Values at endpoints:
We need to check whether f(-8) = f(8). Evaluating the function at the endpoints:
f(-8) = (-8)^(2/3) - 2 = 4 - 2 = 2
f(8) = (8)^(2/3) - 2 = 4 - 2 = 2
Since f(-8) = f(8) = 2, the values at the endpoints are equal.
Based on the above analysis, we can conclude that all the conditions for applying Rolle's Theorem are satisfied. Therefore, option 1) Yes, Rolle's Theorem can be applied is the correct answer.