Increasing and Decreasing Functions

Increasing and decreasing on an interval. Let $f$ be a real function defined on an interval $I$. Then $f$ is said to be:

• increasing on $I$ if for every $x_1, x_2 \in I$, $x_1 < x_2 \implies f(x_1) \le f(x_2)$;
• strictly increasing on $I$ if $x_1 < x_2 \implies f(x_1) < f(x_2)$ (the graph rises continuously, never flat);
• decreasing on $I$ if $x_1 < x_2 \implies f(x_1) \ge f(x_2)$, and strictly decreasing if the inequality is strict.

A function may also be increasing (or decreasing) at a point $x_0$: there is some open interval containing $x_0$ throughout which $f$ is increasing (respectively decreasing) -- a local, rather than a whole-domain, description.

The sign of $f'$ decides monotonicity. Geometrically, $f'(x)$ is the slope of the tangent to $y = f(x)$ at $x$. If this slope is positive throughout an interval, the curve keeps climbing left to right across it; if the slope is negative throughout, the curve keeps falling. This geometric picture is made precise via the Mean Value Theorem (for $x_1 < x_2$ in $I$, there is some $c$ between them with $f(x_2) - f(x_1) = f'(c)(x_2 - x_1)$) into the working test used throughout this chapter:

Method: finding intervals of increase and decrease. For a differentiable function on $\mathbf{R}$ (or on its natural domain):

1. Compute $f'(x)$.
2. Solve $f'(x) = 0$ to find the critical points, which split the domain into open intervals.
3. Test the sign of $f'(x)$ on each interval (one convenient test value per interval suffices, since $f'$ -- typically a polynomial with these critical points as its only real roots -- cannot change sign except at a root).
4. Report each interval as increasing (where $f' > 0$) or decreasing (where $f' < 0$).

This sign-chart method is exactly what Example 4 and every question in Exercise: Increasing and Decreasing Functions apply; see also Exercise: Maxima and Minima, where the same critical points additionally classify local extrema.