Monotonicity and Extrema of Functions
An Algebraic Approach (No Derivatives Required)
Why this matters: Understanding when a function goes up, goes down, or reaches a peak/valley is one of the most powerful ideas in algebra. In this lesson, every argument is purely algebraic — no calculus needed. You will learn to prove monotonicity from the definition, find extrema by completing the square and by the AM–GM inequality, and use monotonicity to solve equations and inequalities.
1. Definitions of Monotonicity
Core Definitions
Let $f$ be a function defined on a set $X$, and let $I \subseteq X$ be an interval.
| Term | Condition (for all $x_1, x_2 \in I$ with $x_1 < x_2$) | Type |
|---|---|---|
| Strictly increasing | $f(x_1) < f(x_2)$ | Strict |
| Strictly decreasing | $f(x_1) > f(x_2)$ | Strict |
| Non-decreasing (weakly increasing) | $f(x_1) \le f(x_2)$ | Non-strict |
| Non-increasing (weakly decreasing) | $f(x_1) \ge f(x_2)$ | Non-strict |
A function is called monotonic (or monotone) on $I$ if it satisfies one of these four conditions on the entire interval $I$.
Strict vs. Non-strict: A strictly increasing function never takes the same value twice — every step to the right produces a strictly higher output. A non-decreasing function may have "flat" segments where $f(x_1) = f(x_2)$ even though $x_1 < x_2$. For example, the floor function $f(x) = \lfloor x \rfloor$ is non-decreasing but not strictly increasing.
Monotonicity on the whole domain vs. on subintervals: A function can be monotonic on its entire domain (e.g., $f(x) = 2x + 3$ is strictly increasing on all of $\mathbb{R}$), or it may be monotonic only on certain subintervals. For instance, $f(x) = x^2$ is strictly decreasing on $(-\infty, 0]$ and strictly increasing on $[0, +\infty)$, but it is not monotonic on all of $\mathbb{R}$.