For the following three (03) items:
Consider the function f(x) = x|x|

Consider the following statements:

I. The function is increasing in the interval (-∞,∞),

II. The function is differentiable at x = 0

Which of the statements given above is/are correct?

For the following two (02) items:
Consider the function
$f(x)=\begin{cases}4(5^{x}),&x<0\\ 8k+x,&x\ge 0\end{cases}$

If the function is continuous, then what is the value of k?

For the following two (02) items:
Let the function f(x) = |x - 3| + |x - 4| be defined on the interval [0, 5].

Consider the following statements:

I. The function is differentiable at x = 3

II. The function is differentiable at x = 4

Which of the statements given above is/are correct?

Consider the following for the two (02) items that follow:

$\text{Let } f(x)= \begin{cases} x^3, & x^2 < 1 \\ x^2, & x^2 \ge 1 \end{cases} \\$

What is  $\lim_{x \to 0} f'(x)$ equal to?