Limit Continuity and Differentiability 7 Question 7
7. Let $f:(-1,1) \rightarrow R$ be a function defined by $f(x)=\max {-x \mid,-\sqrt{1-x^{2}} }$. If $K$ be the set of all points at which $f$ is not differentiable, then $K$ has exactly
(2019 Main, 10 Jan II)
(a) three elements
(b) five elements
(c) two elements
(d) one element
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Answer:
Correct Answer: 7. (b)
Solution:
- We have, $x \log _e\left(\log _e x\right)-x^{2}+y^{2}=4$, which can be written as
$$ y^{2}=4+x^{2}-x \log _e\left(\log _e x\right) $$
Now, differentiating Eq. (i) w.r.t. $x$, we get
$$ 2 y \frac{d y}{d x}=2 x-x \frac{1}{\log _e x} \cdot \frac{1}{x}-1 \cdot \log _e\left(\log _e x\right) $$
[by using product rule of derivative]
$$ \Rightarrow \quad \frac{d y}{d x}=\frac{2 x-\frac{1}{\log _e x}-\log _e\left(\log _e x\right)}{2 y} $$
Now, at $x=e, y^{2}=4+e^{2}-e \log _e\left(\log _e e\right)$
$$ \begin{aligned} & =4+e^{2}-e \log _e(1)=4+e^{2}-0 \\ & =e^{2}+4 \\ & \Rightarrow \quad y=\sqrt{e^{2}+4} \\ & \frac{d y}{d x}=\frac{2 e-1-0}{2 \sqrt{e^{2}+4}}=\frac{2 e-1}{2 \sqrt{e^{2}+4}} \quad \text { [using Eq. (ii)] } \end{aligned} $$
