Q14. State whether (a) x, (b) x³, (c) |x| are increasing.

Since $f'(x) = 1 > 0$ for all real values of $x$, the function $f(x) = x$ is always increasing.

$$\boxed{\text{Yes, } x \text{ is increasing.}}$$

Since $x^2 \ge 0$ for all real values of $x$, $3x^2 \ge 0$ for all $x$. The derivative is zero only at $x=0$, but it is positive for all other $x$. Therefore, the function $f(x) = x^3$ is always increasing.

$$\boxed{\text{Yes, } x^3 \text{ is increasing.}}$$

$$f(x) = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}$$

The derivative is:

$$f'(x) = \begin{cases} 1 & \text{if } x > 0 \\ -1 & \text{if } x < 0 \end{cases}$$

For $x > 0$, $f'(x) = 1 > 0$, so $f(x)$ is increasing.
For $x < 0$, $f'(x) = -1 < 0$, so $f(x)$ is decreasing.
Since the function is decreasing for $x < 0$, it is not an increasing function over its entire domain.

$$\boxed{\text{No, } |x| \text{ is not increasing.}}$$