PolyU AMA1120 Lec 1 Monotonicity, Open and Closed Intervals

Monotonicity

Let $f(x)$ be differentiable on an interval $I$:

• If $f'(x) \geq 0$ for all $x \in I$, then $f$ is increasing on $I$.
• If $f'(x) \leq 0$ for all $x \in I$, then $f$ is decreasing on $I$.
• If $f'(x) > 0$ for all $x \in I$, then $f$ is strictly increasing on $I$.
• If $f'(x) < 0$ for all $x \in I$, then $f$ is strictly decreasing on $I$.

Open and Closed Intervals

• An open interval does not include any endpoint, and is indicated with parentheses: $(a, b)$.
• A closed interval includes all its endpoints and is denoted with square brackets: $[a, b]$.