Applications Of Derivatives Question 280
Question: Let $ f(x)={ \begin{aligned} & {x^{\alpha }}\ln x,x>0 \\ & 0,x=0 \\ \end{aligned} } $ , Rolle-s theorem is applicable to f for $ x\in
[0,1] $ , if $ \alpha = $ [IIT Screening 2004]
Options:
A) - 2
B) - 1
C) 0
D) $ \frac{1}{2} $
Show Answer
Answer:
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.
BETA
