JEE Main & Advanced
Mathematics
Applications of Derivatives
Question Bank
Critical Thinking
question_answer
Let \[f(x)=\left\{ \begin{align} & {{x}^{\alpha }}\ln x,x>0 \\ & 0,\,\,\,\,\,\,\,\,\,\,\,\,x=0 \\ \end{align} \right\}\], Rolle?s theorem is applicable to f for \[x\in [0,1]\], if \[\alpha =\] [IIT Screening 2004]
A)- 2
B)- 1
C)0
D)\[\frac{1}{2}\]
Correct Answer:
D
Solution :
For Rolle?s theorem to be applicable to f, for \[x\in [0,\,1]\], we should have (i) \[f(1)=f(0)\],
(ii) f is continuous for \[x\in [0,\,1]\] and f is differentiable for \[x\in (0,\,1)\]
From (i), \[f(1)=0\], which is true.
From (ii), \[0=f(0)=f({{0}_{+}})=\underset{x\to {{0}_{+}}}{\mathop{\lim }}\,{{x}^{\alpha }}\ln x\] Which is true only for positive values of \[\alpha \], thus is correct.