AP® Calculus AB · Unit 5 · Topic 5.3
Determining Intervals on Which a Function Is Increasing or Decreasing
A function is increasing where its derivative is positive and decreasing where its derivative is negative. A sign chart of f' splits the x-axis into these rising and falling stretches.
The derivative f'(x) is the slope of the tangent line at x. If that slope is positive, the curve is climbing as x increases. If it's negative, the curve is falling. So instead of staring at a graph of f, you can study the sign of f' and know the direction of f without ever plotting a point. This works because slope measures instantaneous rate of change, and a positive rate of change means the output is growing.
To build a sign chart, find all critical points: values where f'(x) = 0 or f' is undefined. These points are the only places the sign of f' can switch, so they break the number line into separate intervals. Pick a test value inside each interval, plug it into f', and record whether the result is positive or negative. Positive means f is increasing on that stretch, negative means decreasing. For f(x) = x3 - 6x2 + 9x, f'(x) = 3x2 - 12x + 9 = 3(x-1)(x-3). Testing around x = 1 and x = 3 shows f increasing on (-infinity, 1), decreasing on (1, 3), and increasing again on (3, infinity).
This sign chart is the engine behind the first derivative test. Where f' switches from positive to negative, the curve stops rising and starts falling, so that critical point is a relative maximum. Where f' switches from negative to positive, you get a relative minimum. If the sign doesn't change at all, there's no extremum there, just a pause. This same chart, read through the lens of concavity instead of direction, is exactly the tool you reuse in the next topic with f'' to find where the curve bends.
Key terms
- Critical point
- A value in the domain of f where f'(x) = 0 or f' is undefined.
- Sign chart
- A number line marked with critical points and the sign of f' on each resulting interval.
- Increasing interval
- A stretch of x-values where f' is positive and the function's output rises as x increases.
- First derivative test
- A method that classifies a critical point as a max, min, or neither by checking how the sign of f' changes around it.
Watch it explained
- How it worksHow does a sign chart for f prime split the number line into rising and falling stretches?
- Worked exampleFind the intervals where f(x) = x3 - 6x2 + 9x rises and where it falls.
- Common mistakeWhy isn't a function with a zero derivative at one point flat all around it?
- Ties togetherHow does a sign chart of the first derivative become the first derivative test for relative extrema?
Check yourself
End of topic.