What Is the Interval of Increase? A Clear Calculus Guide
The interval of increase is the set of x-values over which a function’s output rises as x rises—and on a differentiable function, you find it by solving f′(x) > 0. In other words, the interval of increase is where the graph climbs from left to right, so pinpointing “what is the interval of increase” always comes down to one derivative inequality.
A function is increasing on an interval when, for any two points a and b in that interval with a < b, you get f(a) < f(b). This “as x goes up, y goes up” behavior is what the phrase “interval of increase” describes, and it is always reported as an interval of x-values, such as (0, 3) or (−∞, 2), never as a single x-value.
Definition of Interval of Increase
An interval of increase is an open interval of x-values where the function is strictly rising—larger inputs produce larger outputs. Formally, f is increasing on an interval if for every pair a < b in the interval, f(a) < f(b). When this holds across the function’s entire domain, the function itself is called an increasing function; an interval of increase is simply the piece, or pieces, of the domain where that behavior occurs.
It helps to picture the graph being traced from left to right. Wherever the curve goes uphill, you are inside an interval of increase. Wherever it goes downhill, you are in an interval of decrease. Flat spots—horizontal tangents or constant stretches—belong to neither.
For a smooth, differentiable function, “going uphill” has a precise signature: the slope is positive. That slope is the derivative f′(x), so the interval of increase is exactly where f′(x) > 0. Note that open intervals are used by convention: a single point where the slope is momentarily zero does not create or break an increasing stretch. This derivative-based view is the standard one taught in introductory calculus, as covered by Khan Academy.
How to Find It
The method is a short, repeatable procedure. It works for polynomials, rational functions, and most functions you meet in first-year calculus.
The four-step method
- Find the derivative. Compute f′(x) from f(x).
- Set the derivative greater than zero. Solve the inequality f′(x) > 0.
- Solve the inequality. Factor or otherwise analyze f′(x), and find the critical x-values where f′(x) = 0 or is undefined.
- Identify the open intervals. The solution set is your interval of increase—often expressed as a union of intervals.
Watch the domain
For rational functions, f′ may be undefined at points excluded from the domain, and those points must be removed from any interval. Additionally, the derivative’s sign can only change at critical points (where f′ = 0 or is undefined), so test one number from each region between them.
Testing a single sample point per region is enough. If f′ > 0 there, the whole region is increasing. This sign-testing approach is laid out carefully in Paul’s Online Math Notes.
Worked Example
Let f(x) = x³ − 3x. Then f′(x) = 3x² − 3 = 3(x − 1)(x + 1). Set 3(x − 1)(x + 1) > 0. The critical points are x = −1 and x = 1, splitting the line into three regions. Test each:
- x = −2: f′ = 3(4) − 3 = 9 > 0 → increasing
- x = 0: f′ = 0 − 3 = −3 < 0 → decreasing
- x = 2: f′ = 12 − 3 = 9 > 0 → increasing
So f increases on (−∞, −1) and (1, ∞), and decreases on (−1, 1). For more on derivative rules, see MIT OpenCourseWare.
Intervals of Increase vs Decrease
Increasing and decreasing intervals are mirror images, distinguished only by the sign of the derivative. A sign chart organizes both at once: mark each critical point on a number line, then label each region “+” where f′ > 0 (increasing) or “−” where f′ < 0 (decreasing).
| Derivative sign | Behavior | Interval type |
|---|---|---|
| f′(x) > 0 | Rising left to right | Interval of increase |
| f′(x) < 0 | Falling left to right | Interval of decrease |
| f′(x) = 0 | Horizontal tangent (single point) | Critical point, not its own interval |
A few common mistakes are worth flagging. First, using closed brackets to include critical points—use open intervals, since the point of a zero slope is neither rising nor falling. Second, forgetting points where the derivative is undefined (like a vertical asymptote); these split intervals just as zeros do. Third, testing too few points or skipping regions. Fourth, confusing the y-values of a function with its intervals—intervals are always reported in x.
Because a differentiable function can only switch direction at a critical point, its intervals of increase and decrease always alternate along the number line. Working out both in one sign chart is faster and less error-prone than handling them separately.
Key Takeaways
- An interval of increase is the open set of x-values where a function’s outputs rise as x rises.
- For differentiable functions, find it by solving f′(x) > 0.
- Method: take the derivative, set it greater than zero, find critical points, and test a sample point in each region.
- Report intervals as open intervals, and exclude points outside the domain, especially for rational functions.
- Increasing intervals (f′ > 0) and decreasing intervals (f′ < 0) alternate and are best tracked with a single sign chart.
