Let $f(x) = \begin{cases} x^{3/5} & \text{if } x \le 1 \\ -(x - 2)^3 & \text{if } x > 1 \end{cases}$. Then the number of critical points on the graph of the function is:

Match the following:
In the following,$[x]$ denotes the greatest integer less than or equal to $x$.

$(a)$ $x|x|$	$(i)$ continuous in $(-1, 1)$
$(b)$ $\sqrt{|x|}$	$(ii)$ differentiable in $(-1, 1)$
$(c)$ $x+[x]$	$(iii)$ strictly increasing in $(-1, 1)$
$(d)$ $|x-1|+|x+1|$	$(iv)$ not differentiable at,at least one point in $(-1, 1)$

In the following $[x]$ denotes the greatest integer less than or equal to $x$. Match the functions in Column $I$ with the properties in Column $II$.

Column $I$	Column $II$
$(A)$ $f(x) = x|x|$	$(p)$ continuous in $(-1, 1)$
$(B)$ $f(x) = \sqrt{|x|}$	$(q)$ differentiable in $(-1, 1)$
$(C)$ $f(x) = x + [x]$	$(r)$ strictly increasing in $(-1, 1)$
$(D)$ $f(x) = |x - 1| + |x + 1|$	$(s)$ not differentiable at least at one point in $(-1, 1)$

Let $f: (-\infty, \infty) - \{0\} \rightarrow R$ be a differentiable function such that $f^{\prime}(1) = \lim_{a \rightarrow \infty} a^2 f\left(\frac{1}{a}\right)$. Then $\lim_{a \rightarrow \infty} \left[ \frac{a(a+1)}{2} \tan^{-1}\left(\frac{1}{a}\right) + a^2 - 2 \log_e a \right]$ is equal to

Let $f(x)=x^2+a x+b$,where $a, b \in R$. If $f(x)=0$ has all its roots imaginary,then the roots of $f(x)+f^{\prime}(x)+f^{\prime \prime}(x)=0$ are

Consider the function $f:(-\infty, \infty) \rightarrow(-\infty, \infty)$ defined by $f(x)=\frac{x^2-a x+1}{x^2+a x+1}, 0 < a < 2 .$
$1.$ 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 $(0, \infty)$
$(C)$ $g^{\prime}(x)$ changes sign on both $(-\infty, 0)$ and $(0, \infty)$
$(D)$ $g^{\prime}(x)$ does not change sign on $(-\infty, \infty)$
Give the answer for questions $1, 2$ and $3.$