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Q.
In which one of the following intervals Rolle's theorem hold(s) good for $y=x^2 \sin \frac{1}{x}+x^3 \cos \frac{1}{2 x}$.
Application of Derivatives
Solution:
We have $f ( x )= x ^2 \sin \frac{1}{ x }+ x ^3 \cos \frac{1}{2 x } ; f ^{\prime}( x )$ has opposite signs in $\left[\frac{1}{2 \pi}, \frac{1}{\pi}\right]$ $\Rightarrow f ^{\prime}( x )=0$ atleast once
Clearly $f(x)$ is continuous as well as differentiable in $\left[\frac{1}{3 \pi}, \frac{1}{\pi}\right]$.
Also $f \left(\frac{1}{3 \pi}\right)=0= f \left(\frac{1}{\pi}\right) $ (Only in this interval this will be true)
So, $f ( x )$ satisfies Rolle's theorem.
Hence there exist some $c \in\left(\frac{1}{3 \pi}, \frac{1}{\pi}\right)$ such that $f ^{\prime}( c )=0$.