Limit Continuity and Differentiability 7 Question 10
10. Let $S=\left(t \in \mathbf{R}: f(x) \quad=|x-\pi| \cdot\left(e^{|x|}-1\right) \sin |x|\right.$ is not differentiable at $t}$.Then, the set $S$ is equal to (2018 Main)
(a) $\varphi$ (an empty set)
(b) ${0}$
(c) ${\pi}$
(d) ${0, \pi}$
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Answer:
Correct Answer: 10. (d)
Solution:
- Let $y=\tan ^{-1} \frac{6 x \sqrt{x}}{1-9 x^{3}}=\tan ^{-1} \frac{2 \cdot\left(3 x^{3 / 2}\right)}{1-\left(3 x^{3 / 2}\right)^{2}}$
$$ \begin{aligned} & =2 \tan ^{-1}\left(3 x^{3 / 2}\right) \because 2 \tan ^{-1} x=\tan ^{-1} \frac{2 x}{1-x^{2}} \\ \therefore \quad \frac{d y}{d x} & =2 \cdot \frac{1}{1+\left(3 x^{3 / 2}\right)^{2}} \cdot 3 \times \frac{3}{2}(x)^{1 / 2}=\frac{9}{1+9 x^{3}} \cdot \sqrt{x} \\ \therefore \quad g(x) & =\frac{9}{1+9 x^{3}} \end{aligned} $$
