1. Tardigrade
  2. Question
  3. Mathematics
  4. Let g(x)= begincases(π/2)-2 tan -1 f(x) f(x) ∈(-1,1) -(π/2)-2 tan -1 f(x) f(x) ∈(-∞,-1) text where f(x)= begincasesx+1 x ≤ 0 1-x2 0< x< 2 x-5 x ≥ 2 endcases (3 π/2)-2 tan -1 f(x) f(x) ∈(1, ∞) endcases Column I Column II A Values of x for which derivative of g ( x ) w.r.t. f ( x ) is -(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

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Q. Let $g(x)=\begin{cases}\frac{\pi}{2}-2 \tan ^{-1} f(x) & f(x) \in(-1,1) \\ -\frac{\pi}{2}-2 \tan ^{-1} f(x) & f(x) \in(-\infty,-1) \text { where } f(x)=\begin{cases}x+1 & x \leq 0 \\ 1-x^2 & 0< x< 2 \\ x-5 & x \geq 2\end{cases} \\ \frac{3 \pi}{2}-2 \tan ^{-1} f(x) & f(x) \in(1, \infty)\end{cases}$
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

Application of Derivatives

Solution:

Let $ g(x)=\begin{cases}\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{cases}$
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(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{-2 f ^{\prime}( x )}{1+ f ^2( x )}<0 \Rightarrow f ^{\prime}( x )>0 \Rightarrow x =-6,10 \Rightarrow P , S$