Limit Continuity and Differentiability 7 Question 4
4. Let $S$ be the set of all points in $(-\pi, \pi)$ at which the function, $f(x)=\min {\sin x, \cos x}$ is not differentiable. Then, $S$ is a subset of which of the following?
(a) $-\frac{\pi}{4}, 0, \frac{\pi}{4}$
(b) $-\frac{\pi}{2},-\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2}$
(c) $-\frac{3 \pi}{4},-\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{\pi}{4}$
(d) $-\frac{3 \pi}{4},-\frac{\pi}{2}, \frac{\pi}{2}, \frac{3 \pi}{4}$
(2019 Main, 12 Jan I)
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Answer:
Correct Answer: 4. (c)
Solution:
- Let $y=f(f(f(x)))+(f(x))^{2}$
On differentiating both sides w.r.t. $x$, we get
$$ \frac{d y}{d x}=f^{\prime}(f(f(x))) \cdot f^{\prime}(f(x)) \cdot f^{\prime}(x)+2 f(x) f^{\prime}(x) $$
[by chain rule]
$$ \begin{aligned} & \text { So, }\left.\frac{d y}{d x}\right| _{\text {at } x=1}=f^{\prime}(f(f(1))) \cdot f^{\prime}(f(1)) \cdot f^{\prime}(1)+2 f(1) f^{\prime}(1) \\ & \left.\therefore \quad \frac{d y}{d x}\right| _{x=1}=f^{\prime}(f(1)) \cdot f^{\prime}(1) \cdot(3)+2(1)(3) \\ & =f^{\prime}(1) \cdot(3) \cdot(3)+6 \\ & =(3 \times 9)+6=27+6=33 \end{aligned} $$
