Increasing and Decreasing Functions
Calculus & Advanced MathA function is increasing where its graph rises from left to right and decreasing where it falls, determined by the sign of the first derivative.
Formula
f'(x) > 0: \text{ increasing; } f'(x) < 0: \text{ decreasing}
Definition
A function is increasing on an interval if the output goes up as you move right. It is decreasing if the output goes down as you move right.
Example
$f(x) = x^2$ is decreasing for $x < 0$ (going downhill toward $0$) and increasing for $x > 0$ (going uphill away from $0$).
Key Insight
The sign of the slope (derivative) tells you the direction: positive slope means going up, negative slope means going down.
Definition
$f$ is (strictly) increasing on $[a, b]$ if $a < b$ implies $f(a) < f(b)$. If $f$ is differentiable, then $f' > 0$ on $(a, b)$ implies $f$ is increasing there; $f' < 0$ implies decreasing. Critical points separate increasing and decreasing intervals.
Example
$f(x) = x^3 - 3x$: $f'(x) = 3x^2 - 3 = 3(x-1)(x+1)$. $f' > 0$ on $(-\infty,-1)$ and $(1,\infty)$: increasing. $f' < 0$ on $(-1,1)$: decreasing.
Key Insight
Identifying increasing and decreasing intervals is a key step in graphing functions and solving optimization problems.
Definition
Formally, $f$ is non-decreasing if $x < y$ implies $f(x) \le f(y)$, and strictly increasing if the inequality is strict. A function with $f' \ge 0$ (and $f'$ not identically zero on any interval) is strictly increasing. Monotone functions have at most countably many discontinuities and are measurable.
Example
The inverse function theorem: if $f$ is continuously differentiable with $f'(a) \neq 0$, then $f$ is locally strictly monotone near $a$, guaranteeing the existence of a local inverse function.
Key Insight
Monotone functions are central to real analysis (they are Riemann integrable), measure theory (distribution functions of measures), and order theory (monotone mappings between posets).