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

The function $f(x) = \begin{cases} \frac{\pi}{4} + \tan^{-1} x, & |x| \leq 1 \\ \frac{1}{2}(|x|-1), & |x| > 1 \end{cases}$ is:

Consider the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = \begin{cases} (2 - \sin(\frac{1}{x}))|x|, & x \neq 0 \\ 0, & x = 0 \end{cases}$. Then $f$ is

Let $f:[0,3] \rightarrow R$ be defined by $f(x)=\min \{x-[x], 1+[x]-x\}$,where $[x]$ is the greatest integer less than or equal to $x$. Let $P$ denote the set containing all $x \in[0,3]$ where $f$ is discontinuous,and $Q$ denote the set containing all $x \in(0,3)$ where $f$ is not differentiable. Then the sum of number of elements in $P$ and $Q$ is equal to $......$

$f(x)$ is a differentiable function and given $f^{\prime}(2)=6$ and $f^{\prime}(1)=4$,then $L=\lim _{h \rightarrow 0} \frac{f\left(2+2 h+h^2\right)-f(2)}{f\left(1+h-h^2\right)-f(1)}$

Let $h$ be a twice continuously differentiable positive function on an open interval $J.$ Let $g(x) = \ln(h(x))$ for each $x \in J$. Suppose $(h'(x))^2 > h''(x) h(x)$ for each $x \in J$. Then