Differentiability

Step 1: Conditions for Rolle's Theorem
Rolle's Theorem states that if a function is: 1. Continuous on the closed interval , 2. Differentiable on the open interval , 3. , then there exists such that .

Step 2: Checking Continuity
For to be continuous at , we check: Using the standard limit result, Since , the function is continuous at if .

Step 3: Checking Differentiability
Differentiate for : For to be differentiable at , the right-hand derivative should exist: Using limits, is differentiable at if .

Step 4: Verifying
Since , the condition holds.

Step 5: Conclusion
For Rolle's theorem to hold, we need to satisfy all conditions. Thus, the correct answer is: