Inverse Trigonometric Functions — Maths STD 12 Science — Question

$1.$ Which of the following is true?

$(A)$ $(2+a)^2 f^{\prime \prime}(1)+(2-a)^2 f^{\prime \prime}(-1)=0$

$(B)$ $(2-a)^2 f^{\prime}(1)-(2+a)^2 f^{\prime \prime}(-1)=0$

$(C)$ $f^{\prime}(1) f^{\prime}(-1)=(2-a)^2$

$(D)$ $f^{\prime}(1) f^{\prime}(-1)=-(2+a)^2$

$2.$ Which of the following is true?

$(A)$ $f(x)$ is decreasing on $(-1,1)$ and has a local minimum at $x=1$

$(B)$ $f(x)$ is increasing on $(-1,1)$ and has a local maximum at $x=1$

$(C)$ $f(x)$ is increasing on $(-1,1)$ but has neither a local maximum nor a local minimum at $x=1$

$(D)$ $f(x)$ is decreasing on $(-1,1)$ but has neither a local maximum nor a local minimum at $x=1$

$3.$ Let $g(x)=\int_0^{e^x} \frac{f^{\prime}(t)}{1+t^2} d t$ which of the following is true?

$(A)$ $g^{\prime}(x)$ is positive on $(-\infty, 0)$ and negative on $(0, \infty)$

$(B)$ $g^{\prime}(x)$ is negative on $(-\infty, 0)$ and positive on $(0, \infty)$

$(C)$ $\mathrm{g}^{\prime}(\mathrm{x})$ changes sign on both $(-\infty, 0)$ and $(0, \infty)$

$(D)$ $g^{\prime}(x)$ does not change sign on $(-\infty, \infty)$

Give the answer question $1,2$ and $3.$

1. $\frac{1}{2}\log(-1)$
   
2. $\log(-1)$
   
3. $\log3$
   
4. $\log\sqrt{3}$
   

1. $9\ \text{sec}.$
   
2. $\frac{5}{3}\ \text{sec}.$
   
3. $\frac{3}{5}\ \text{sec}.$
   
4. $2\ \text{sec}.$