Consider $f(x) = \begin{cases} \tan^{-1}(\frac{\alpha x + \beta}{\gamma}) & x \in (0, \frac{1}{2}) \\ 0 & x = \frac{1}{2} \\ \ln(\beta x^2 + 2) & x \in (\frac{1}{2}, 1) \end{cases}$. If $f(x)$ is continuous and differentiable in its domain,then the value of $\alpha + \beta + \gamma$ is:

Let $y(x) = (1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})$. Then $y'(x) - y''(x)$ at $x = -1$ is equal to:

Let $f:R \to R$ be a continuous function defined by $f(x) = \frac{1}{e^x + 2e^{-x}}$.
Statement-$1$: $f(c) = \frac{1}{3}$ for some $c \in R$.
Statement-$2$: $0 < f(x) < \frac{1}{2\sqrt{2}}$ for all $x \in R$.

The number of points,at which the function $f(x) = \max\{6x, 2+3x^2\} + |x-1| |\cos(x^2 - 1/4)|, x \in (-\pi, \pi)$,is not differentiable,is ————

Let $f:\left[-\frac{1}{2}, 2\right] \rightarrow R$ and $g:\left[-\frac{1}{2}, 2\right] \rightarrow R$ be functions defined by $f(x)=\left[x^2-3\right]$ and $g(x)=|x| f(x)+|4 x-7| f(x)$,where $[y]$ denotes the greatest integer less than or equal to $y$ for $y \in R$. Then
$(A)$ $f$ is discontinuous exactly at three points in $\left[-\frac{1}{2}, 2\right]$
$(B)$ $f$ is discontinuous exactly at four points in $\left[-\frac{1}{2}, 2\right]$
$(C)$ $g$ is $NOT$ differentiable exactly at four points in $\left(-\frac{1}{2}, 2\right)$
$(D)$ $g$ is $NOT$ differentiable exactly at five points in $\left(-\frac{1}{2}, 2\right)$

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