To determine if Rolle's theorem can be applied to the function $f(x) = x^{2/3} - 6$ on the interval $[-27, 27]$, we need to check if the function meets the three criteria for Rolle's theorem:

1. Continuity on the closed interval $[a, b]$.
2. Differentiability on the open interval $(a, b)$.
3. Equal values at the endpoints, i.e., $f(a) = f(b)$.

Step 1: Check Continuity on $[-27, 27]$

The function $f(x) = x^{2/3} - 6$ is continuous for all real values of $x$ because it is a transformation of the continuous function $x^{2/3}$. Therefore, $f(x)$ is continuous on the interval $[-27, 27]$.

Step 2: Check Differentiability on $(-27, 27)$

To check differentiability, we need the derivative of $f(x)$: $f(x) = x^{2/3} - 6$ $f'(x) = \frac{2}{3} x^{-1/3} = \frac{2}{3} \cdot \frac{1}{x^{1/3}}$ The derivative $f'(x) = \frac{2}{3} \cdot \frac{1}{x^{1/3}}$ is undefined at $x = 0$ (since dividing by $x^{1/3}$ when $x = 0$ is not possible). Thus, $f(x)$ is not differentiable on the entire interval $(-27, 27)$.

Step 3: Check $f(-27) = f(27)$

Calculate the values at the endpoints: $f(-27) = (-27)^{2/3} - 6 = 9 - 6 = 3$ $f(27) = (27)^{2/3} - 6 = 9 - 6 = 3$ Since $f(-27) = f(27) = 3$, the third condition is met.

Conclusion

Rolle's theorem cannot be applied because $f(x)$ is not differentiable on $(-27, 27)$ due to the point at $x = 0$.

Answer

The correct choices are:

• No, because $f$ is not differentiable on the open interval $(a, b)$.

For the value of $c$:

• Since Rolle's theorem cannot be applied, enter NA.

Would you like further details or have any questions about this problem?

1. What is Rolle's theorem and how does it apply to differentiable functions?
2. How do you determine the continuity of a function on a closed interval?
3. What makes a function not differentiable at certain points?
4. Why is $x^{2/3}$ not differentiable at $x = 0$?
5. Can there be cases where Rolle's theorem applies even if $f'(c) = 0$ at a point other than $c = 0$?

Tip: For functions involving fractional exponents, check differentiability at zero carefully, as these functions often have issues at that point.