Increasing and Decreasing Functions

Once we can differentiate a function, the derivative becomes a powerful tool for describing how the function behaves — whether its values are rising or falling, where it turns, and where it reaches a largest or smallest value. These questions matter directly in commerce, where a cost, a revenue or a profit is a function of the quantity produced, and a manager wants to know exactly where that quantity should be set.

The first and most basic behaviour is whether a function is increasing or decreasing over an interval.

A function $f$ is said to be increasing on an interval $I$ if, whenever $x_1 < x_2$ in $I$, we have $f(x_1) \le f(x_2)$ — larger inputs give larger (or equal) outputs. It is strictly increasing if $x_1 < x_2 \Rightarrow f(x_1) < f(x_2)$. Similarly $f$ is decreasing on $I$ if $x_1 < x_2 \Rightarrow f(x_1) \ge f(x_2)$, and strictly decreasing if the inequality is strict.

The derivative test. The sign of the derivative $f'(x)$ tells us the direction of change, because $f'(x)$ is the slope of the tangent — the instantaneous rate at which $f$ is changing. On an interval $I$ where $f$ is differentiable:

• if $f'(x) > 0$ for every $x$ in $I$, then $f$ is strictly increasing on $I$;
• if $f'(x) < 0$ for every $x$ in $I$, then $f$ is strictly decreasing on $I$;
• if $f'(x) = 0$ throughout $I$, then $f$ is constant on $I$.

A positive slope means the curve rises as we move to the right; a negative slope means it falls.

Method for finding the intervals. To find where a function increases or decreases:

1. Compute $f'(x)$.
2. Solve $f'(x) = 0$ to find the critical points — the values of $x$ where the curve may change direction.
3. These critical points divide the number line into open intervals. Test the sign of $f'(x)$ in each interval (pick any convenient test value inside it).
4. Where $f'(x) > 0$ the function is increasing; where $f'(x) < 0$ it is decreasing.

For example, for $f(x) = x^2$ we have $f'(x) = 2x$, which is negative for $x < 0$ and positive for $x > 0$; so $f$ decreases on $(-\infty, 0)$ and increases on $(0, \infty)$, exactly matching the U-shape of its parabola. A function whose derivative is a perfect square or a sum of squares — for instance $f'(x) = 3(2x-3)^2$ — is never negative, so such a function is increasing on the whole real line.