If f ( x ) = x 3 sin ( 1 x ) , x ? 0 0 , x = 0 , then [2024], Differential Calculus jee mains questions

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Solution

Q.
If $f (x)$ $= {\begin{matrix} x^{3} \sin (\frac{1}{x}), & x \neq 0 \\ 0, & x = 0 \end{matrix},$ then [2024]

1 $f'' (0) = 0$

2 $f'' (0) = 1$

3 $f'' (\frac{2}{π}) = \frac{12 - π^{2}}{2 π}$

4 $f'' (\frac{2}{π}) = \frac{24 - π^{2}}{2 π}$

Ans.
(4)

$f (x)$ $= {\begin{matrix} x^{3} \sin \frac{1}{x}, & x \neq 0 \\ 0, & x = 0 \end{matrix},$

It is a differentiable function.

$∴$ $f' (x)$ $= 3 x^{2} \sin \frac{1}{x} + x^{3} \cos \frac{1}{x} (\frac{- 1}{x^{2}})$

$= 3 x^{2} \sin \frac{1}{x} - x \cos \frac{1}{x}$

$f'' (x)$ $= 6 x \sin \frac{1}{x} - 3 \cos \frac{1}{x} - \cos \frac{1}{x} - \frac{\sin \frac{1}{x}}{x}$

$f'' (\frac{2}{π}) = \frac{6 \times 2}{π} - \frac{π}{2} = \frac{24 - π^{2}}{2 π}$

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