If f(x)= cc x^3 (1 x ), & x 0 0, & x=0 ., then - Vidyadip

MCQ

If $f(x)=\left\{\begin{array}{cc}x^3 \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x=0\end{array}\right.$, then

A
$f^{\prime \prime}(0)=1$
B
$f ^{\prime \prime}\left(\frac{2}{\pi}\right)=\frac{24-\pi^2}{2 \pi}$
C
$f^{\prime \prime}\left(\frac{2}{\pi}\right)=\frac{12-\pi^2}{2 \pi}$
D
$f ^{\prime \prime}(0)=0$

✓

Answer

$f^{\prime}(x)=3 x^2 \sin \left(\frac{1}{x}\right)-x \cos
\left(\frac{1}{x}\right)$
$f^{\prime \prime}(x)=6 x \sin \left(\frac{1}{x}\right)-3 \cos \left(\frac{1}{x}\right)-\cos \left(\frac{1}{x}\right)-\frac{\sin \left(\frac{1}{x}\right)}{x}$
$f^{\prime \prime}\left(\frac{2}{\pi}\right)=\frac{12}{\pi}-\frac{\pi}{2}=\frac{24-\pi^2}{2 \pi}$

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