Question

Let $g(x)=\{\begin{array}{ll}\frac{\pi }{2}-2{\tan }^{-1}f(x)& f(x)\in (-1,1)\\ -\frac{\pi }{2}-2{\tan }^{-1}f(x)& f(x)\in (-\infty ,-1)\text{ where }f(x)=\{\begin{array}{ll}x+1& x\le 0\\ 1-x^2& 0<x<2\\ x-5& x\ge 2\end{array}\\ \frac{3\pi }{2}-2{\tan }^{-1}f(x)& f(x)\in (1,\infty )\end{array}$

Column I		Column II
A	Values of $x$ for which derivative of $g(x)$ w.r.t. $f(x)$ is $-\frac{1}{13}$ is/are	P	-6
B	Values of $x$ for which $f(x)$ has local maximum or local minimum is/are	Q	0
C	Values of $k$ for which $f(x)+k=0$ has 2 positive and one	R	2 negative root is/are
D	Values of $x$ for which $g^{\prime }(x)<0$ is/are	S	10

• A. (A) P, S, (B) Q, R; (C) Q, R; (D) P, P
• B. (A) P, S, (B) Q, R; (C) Q, R; (D) P, Q
• C. (A) P, S, (B) Q, R; (C) Q, R; (D) P, S
• D. (A) P, S, (B) Q, R; (C) Q, R; (D) P, R

Answer 1

Answer: (C)

Let $g(x)=\{\begin{array}{ll}\frac{\pi }{2}-2{\tan }^{-1}f(x)& f(x)\in (-1,1)\\ -\frac{\pi }{2}-2{\tan }^{-1}f(x)& f(x)\in (-\infty ,-1)\\ \frac{3\pi }{2}-2{\tan }^{-1}f(x)& f(x)\in (1,\infty )\end{array}$

(A) $\frac{dg(x)}{df(x)}=-\frac{2}{1+f^2(x)}=-\frac{1}{13}\Rightarrow f(x)=\pm 5\Rightarrow x=-6,10\Rightarrow P,S$
(B) refer to graph of $y=f(x)\Rightarrow Q,R$
(C) $-k\in (-3,1)\Rightarrow k\in (-1,3)\Rightarrow Q,R$
(D) $g^{\prime }(x)=\frac{-2f^{\prime }(x)}{1+f^2(x)}<0\Rightarrow f^{\prime }(x)>0\Rightarrow x=-6,10\Rightarrow P,S$