To determine which functions satisfy the assumptions of Rolle's Theorem on the interval $[-1, 1]$, let's review the requirements:

1. Continuity on the closed interval $[-1, 1]$.
2. Differentiability on the open interval $(-1, 1)$.
3. The function's values at the endpoints must be equal, i.e., $f(-1) = f(1)$.

Let's examine each function in this context:

1. $f(x) = 1 + x^{2/3}$

• Continuity: This function is continuous on $[-1, 1]$ because it involves powers of $x$ that do not cause any discontinuities.
• Differentiability: $f(x)$ is not differentiable at $x = 0$ because $x^{2/3}$ has a non-differentiable point at $x = 0$ (the derivative approaches infinity as $x$ approaches 0).
• Therefore, $f(x)$ does not satisfy Rolle's Theorem.

2. $g(x) = \ln |x|$

• Continuity: $g(x)$ is not continuous on $[-1, 1]$ because $\ln |x|$ is undefined at $x = 0$.
• Since continuity is required for Rolle's Theorem, $g(x)$ does not satisfy Rolle's Theorem.

3. $h(x) = \frac{x^2}{4 - x^2}$

• Continuity: $h(x)$ has vertical asymptotes at $x = \pm 2$, which are outside of $[-1, 1]$, so it is continuous on $[-1, 1]$.
• Differentiability: $h(x)$ is differentiable on $(-1, 1)$.
• Endpoint Values: $h(-1) = \frac{1}{3}$ and $h(1) = \frac{1}{3}$, so $h(-1) = h(1)$.
• Therefore, $h(x)$ satisfies all conditions of Rolle's Theorem.

4. $k(x) = \frac{x}{1 + x^2}$

• Continuity: This function is continuous on $[-1, 1]$.
• Differentiability: It is differentiable on $(-1, 1)$.
• Endpoint Values: $k(-1) = -\frac{1}{2}$ and $k(1) = \frac{1}{2}$, so $k(-1) \neq k(1)$.
• Therefore, $k(x)$ does not satisfy Rolle's Theorem.

Conclusion

The only function that satisfies all the conditions of Rolle's Theorem on $[-1, 1]$ is $h(x)$.

Answer:

The correct choice is:

• h only

Would you like a deeper explanation or have any questions?

Related Questions:

1. What are the necessary conditions for a function to satisfy Rolle's Theorem?
2. How do vertical asymptotes affect the continuity and differentiability of a function?
3. Can a function with a point of non-differentiability ever satisfy Rolle’s Theorem on an interval containing that point?
4. What would happen if the interval endpoints do not yield the same function value in Rolle's Theorem?
5. How does the Mean Value Theorem relate to Rolle's Theorem?

Tip:

When checking if a function satisfies Rolle’s Theorem, always verify differentiability across the entire open interval, as points of non-differentiability disqualify the function from meeting the theorem's conditions.