Prove the many other variants!

Let \(a \lt b.\) Let \(f\) be a function defined on \([a, b].\)
$$\begin{align}
& \text{If \(\forall x \in (a, b), f'(x) \ge 0,\)} \\
& \text{and \(f\) is continuous on \([a, b],\)} \\
& \text{then \(f\) is increasing on \([a, b].\)}
\end{align}$$

Let \(a \lt b.\) Let \(f\) be a function defined on \([a, b].\)
$$\begin{align}
& \text{If \(\forall x \in (a, b), f'(x) \le 0,\)} \\
& \text{and \(f\) is continuous on \([a, b],\)} \\
& \text{then \(f\) is decreasing on \([a, b].\)}
\end{align}$$