The value of k for which the function $f(x) = \begin{cases} \frac{1-\cos 8x}{16x^2}, & \text{if } x \neq 0 \\ k, & \text{if } x = 0 \end{cases}$ is continuous at $x = 0$ is:

$\frac{d}{dx}\left(e^{2\log_e x^3}\right)$ equals

If $f(x) = \begin{cases} \frac{\tan(\pi/4 - x)}{\cot 2x}, & x \neq \pi/4 \\ k, & x = \pi/4 \end{cases}$ is continuous at $x = \pi/4$, then the value of k is

If $x = \frac{a}{1 + t}$ and $y = \frac{a}{(1 + t)^2}$ where $a > 0$ , then $\frac{d^2y}{dx^2}$ at $t = 1$ is

If $f(x) = \begin{cases} ax^2 + b, & x < -1 \\ bx^2 + ax + 4, & x \geq -1 \end{cases}$ is everywhere differentiable, then :

If $f(x) = |x| + |x - 5|$, then which of the following statements are TRUE?

(A) f is a continuous function every where

(B) f is a continuous function except $x = 5$ and $x = 0$

(C) f is a continuous function except $x = 0$ but not differentiable at $x = 5$

(D) f is a continuous function everywhere but not differentiable at $x = 0$ and $x = 5$

Choose the correct answer from the options given below:

If $f(x+y) = f(x) + f(y)$ for all $x, y \in R$ (set of real numbers) and $f(x)$ is continuous at $x = 0$, then identify the correct statement ?