If f(x) = x^alogx and f(0) = 0, then the value of a for which rolle’s theorem is applicable in [0,1] is ?

To determine the value of \( a \) for which Rolle's Theorem is applicable to the function \( f(x) = x^a \log x \) on the interval \([0, 1]\), we need to ensure that the function meets the conditions of the theorem. These conditions are: the function must be continuous on the closed interval \([0, 1]\), differentiable on the open interval \((0, 1)\), and the values of the function at the endpoints must be equal, i.e., \( f(0) = f(1) \).

Analyzing the Function

First, let's evaluate the function at the endpoints:

• At \( x = 1 \):

  We have \( f(1) = 1^a \log(1) = 1 \cdot 0 = 0 \).

• At \( x = 0 \):

  Given that \( f(0) = 0 \), we find \( f(0) = 0 \) as stated.

Thus, both endpoints yield \( f(0) = f(1) = 0 \), satisfying the first condition of Rolle's Theorem.

Continuity and Differentiability

Next, we need to check the continuity and differentiability of \( f(x) \) on the specified intervals. The function \( f(x) = x^a \log x \) is defined for \( x > 0 \). However, we need to consider the behavior as \( x \) approaches 0.

The logarithmic function \( \log x \) approaches \(-\infty\) as \( x \) approaches 0. Therefore, the product \( x^a \log x \) will depend on the value of \( a \):

• If \( a > 0 \):

  As \( x \to 0^+ \), \( x^a \) approaches 0, and \( \log x \) approaches \(-\infty\). The product \( x^a \log x \) approaches 0, making \( f(0) = 0 \) continuous.

• If \( a = 0 \):

  Then \( f(x) = \log x \), which is not defined at \( x = 0 \), thus failing continuity.

• If \( a < 0 \):

  In this case, \( x^a \) approaches \(\infty\) as \( x \to 0^+\), leading to \( f(x) \) being undefined at \( x = 0\), again failing continuity.

Conclusion on the Value of a

From this analysis, we conclude that for \( f(x) \) to be continuous on \([0, 1]\) and differentiable on \((0, 1)\), \( a \) must be greater than 0. Therefore, the value of \( a \) for which Rolle's Theorem is applicable in the interval \([0, 1]\) is:

a > 0