Limits Continuity And Differentiability Question 85
If $ f(x)={x^{\alpha }}\log x $ and $ f(0)=0 $ , then the value of α for which Rolle’s theorem can be applied in [0, 1] is
Options:
A) -2
B) -1
C) 0
D) ½
Show Answer
Answer:
Correct Answer: D
Solution:
For Rolle’s theorem in [a, b], f(a)=f(b), In $ [0,1]\Rightarrow f(0)=f(1)=0 $
$ \because $ the function has to be continuous in [0, 1]
$ \Rightarrow f(0)=\underset{x\to {0^{+}}}{\mathop{\lim }}f(x)=0\Rightarrow \underset{x\to 0^{+}}{\mathop{\lim }}{x^{\alpha }}\log x=0 $
$ \Rightarrow \underset{x\to 0}{\mathop{\lim }}\frac{\log x}{{x^{-\alpha }}}=0 $ Applying L.H. Rule $ \underset{x\to 0}{\mathop{\lim }}\frac{1/x}{-\alpha {x^{-\alpha -1}}}=+\infty $
$ \Rightarrow \underset{x\to 0}{\mathop{\lim }}\frac{-{x^{\alpha }}}{\alpha }=0\Rightarrow \alpha >0 $
BETA
