Increasing and Decreasing Functions
Understanding how functions behave — where they rise, where they fall — is one of the most powerful applications of the derivative in calculus.
🔍 The Big Idea
Imagine walking along a hilly road. Sometimes you're going uphill (the function is increasing), and sometimes you're going downhill (the function is decreasing). Calculus gives us a precise tool — the first derivative — to determine exactly where each behavior occurs.
📐 Formal Definitions
A function $f$ is increasing on an interval $(a, b)$ if:
$$\text{For all } x_1, x_2 \in (a,b), \quad x_1 < x_2 \implies f(x_1) < f(x_2)$$
In words: as $x$ moves to the right, $f(x)$ goes up.
A function $f$ is decreasing on an interval $(a, b)$ if:
$$\text{For all } x_1, x_2 \in (a,b), \quad x_1 < x_2 \implies f(x_1) > f(x_2)$$
In words: as $x$ moves to the right, $f(x)$ goes down.