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

The given function is $f(x) = x|x|$.

This can be redefined as:
$f(x) = x^2$ for $x \ge 0$
$f(x) = -x^2$ for $x < 0$

Differentiating with respect to $x$, we get:
$f'(x) = 2x$ for $x > 0$
$f'(x) = -2x$ for $x < 0$

At $x = 0$, the left-hand derivative is $\lim_{h \to 0^-} \dfrac{-h^2 - 0}{h} = 0$ and the right-hand derivative is $\lim_{h \to 0^+} \dfrac{h^2 - 0}{h} = 0$.

Since the left-hand derivative equals the right-hand derivative, $f(x)$ is differentiable at $x = 0$. Thus, statement II is correct.

Also, $f'(x) = 2|x| \ge 0$ for all $x \in (-\infty, \infty)$. Since $f'(x) > 0$ for all $x \neq 0$ and $f(x)$ is continuous, the function is strictly increasing in $(-\infty, \infty)$. Thus, statement I is correct.

Both statements I and II are correct.

Answer: Both I and II