Let $f:R \to R$ be a continuously differentiable function such that $f(2) = 6$ and $f'(2) = \frac{1}{48}$. If $\int_6^{f(x)} 4t^3 \,dt = (x - 2)g(x)$,then $\lim_{x \to 2} g(x)$ is equal to

If a function $f(x)$ defined by $f(x)=\begin{cases} a e^{x}+b e^{-x}, & -1 \leq x<1 \\ c x^{2}, & 1 \leq x \leq 3 \\ a x^{2}+2 c x, & 3 < x \leq 4 \end{cases}$ is continuous for some $a, b, c \in R$ and $f'(0)+f'(2)=e$,then the value of $a$ is:

Let $f(x) = \begin{cases} \frac{1}{|x|}, & |x| \geqslant 1 \\ ax^2 + b, & |x| < 1 \end{cases}$ be continuous and differentiable everywhere. Then $a$ and $b$ are

If $\alpha$ is the angle of intersection between the curves $y = a^x$ and $y = b^x$,then what is $\tan \alpha$ equal to?

Let $f, g: R \rightarrow R$ be two real-valued functions defined as $f(x)=\begin{cases} -|x+3| & , x < 0 \\ e^{x} & , x \geq 0 \end{cases}$ and $g(x)=\begin{cases} x^{2}+k_{1} x & , x < 0 \\ 4 x+k_{2} & , x \geq 0 \end{cases}$,where $k_{1}$ and $k_{2}$ are real constants. If $(g \circ f)$ is differentiable at $x=0$,then $(g \circ f)(-4)+(g \circ f)(4)$ is equal to

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