Question

If \( f(x) = x^\alpha \log x \) and \( f(0) = 0 \), then the value of \( \alpha \) for which Rolle's theorem can be applied in \( [0, 1] \) is:

• A. -2
• B. -1
• C. 0
• D. \( \frac{1}{2} \)

Hint:

For Rolle’s theorem, the function must be continuous and differentiable on the interval, and have equal values at the endpoints.

Answer 1

The Correct Option is D

Step 1: Apply Rolle’s Theorem.
Rolle's theorem can be applied when the function is continuous, differentiable, and the function values at the endpoints are equal. By solving for \( \alpha \), we find \( \alpha = \frac{1}{2} \).
Thus, the correct answer is (4).