Question

In which one of the following intervals does Rolle's theorem hold good for
y $=x^{2}sin\frac{1}{x}+x^{3}cos\frac{1}{2x}$.

• A. $[\frac{1}{\pi },\frac{2}{\pi }]$
• B. $[\frac{1}{3\pi },\frac{1}{\pi }]$
• C. $[\frac{1}{2\pi },\frac{1}{\pi }]$
• D. $[\frac{1}{\pi },\frac{3}{\pi }]$

Answer 1

The correct option is B $[\frac{1}{3\pi },\frac{1}{\pi }]$

$y=x^{2}sin\frac{1}{x}+x^{3}cos\frac{1}{2x}$.
We will check f(a)=f(b) for each option.
Option A , $[\frac{1}{\pi },\frac{2}{\pi }]$
$f\left(\frac{1}{\pi }\right)=0$ and $f\left(\frac{2}{\pi }\right)=\frac{4\sqrt{2}+4\pi }{{\pi }^3}$
$\Rightarrow$, Rolle's theorem does not holds for $[\frac{1}{\pi },\frac{2}{\pi }]$

Option B , $[\frac{1}{3\pi },\frac{1}{\pi }]$
$f\left(\frac{1}{3\pi }\right)=0$ and $f\left(\frac{1}{\pi }\right)=0$
Hence, Rolle's theorem holds for $[\frac{1}{3\pi },\frac{1}{\pi }]$
It does not holds for other options.
Hence, option B is correct.