If $f(x) = \begin{cases} ax^2 - bx + 2, & x < 3 \\ bx^2 - 3, & x \geq 3 \end{cases}$ is differentiable at every $x \in R$,then the area (in sq units) of the triangle formed by the line $\frac{x}{a} + \frac{y}{b} = 1$ with the coordinate axes is

If $f(x) = \cos x \cos 2x \cos 4x \cos 8x \cos 16x$,then $f'\left( \frac{\pi}{4} \right)$ 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 $......$

The number of real roots of the equation $\log_{e} x + ex = 0$ is

If $y = \operatorname{Tan}^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right) + \operatorname{Tan}^{-1}\left(\frac{7x}{1 - 12x^2}\right)$,then at $x = 0$,$\frac{dy}{dx} = $

If $f(x) = \begin{cases} x, & x \le 0 \\ 0, & x > 0 \end{cases}$ then $f(x)$ at $x = 0$ is