How can the first derivative indicate where a function is increasing or decreasing?
Quick answer
Find the first derivative, then split the function’s domain at points where the derivative is zero or undefined. Check its sign throughout each resulting interval: if it is positive throughout, the function is increasing there; if it is negative throughout, the function is decreasing. A derivative’s value at one point gives slope information, not by itself a conclusion about an entire interval.
The question, the details, the next step.
How do I use the first derivative to tell where a function is increasing or decreasing?
Check the derivative’s sign over intervals, not just at a single point. Where the derivative stays positive, the function is increasing; where it stays negative, the function is decreasing. For a function continuous on a closed interval and differentiable inside it, a positive derivative throughout the interior guarantees that the function is increasing on the interval; a negative derivative throughout guarantees that it is decreasing [S1].
A practical sign check works like this:
- Calculate f′(x).
- Find x-values where f′(x) = 0 or where f′(x) does not exist. These are possible boundaries between intervals.
- Check the sign of f′ throughout each resulting interval.
- State the conclusion interval by interval, taking the function’s domain and any relevant endpoints into account.
Can you work through a polynomial example and show the sign check?
Take f(x) = x² − 4x + 3. Its derivative is f′(x) = 2x − 4. Set the derivative equal to zero: 2x − 4 = 0, so x = 2 divides the real number line into two intervals.
Choose a test value in each interval:
- For x < 2, test x = 0. Then f′(0) = −4, which is negative. The function is decreasing on (−∞, 2).
- For x > 2, test x = 3. Then f′(3) = 2, which is positive. The function is increasing on (2, ∞).
The sign pattern is negative to the left of 2, zero at 2, and positive to the right. So the function changes from decreasing to increasing there. As a check, f(2) = −1, so the point on the graph at x = 2 is (2, −1). The derivative signs establish the interval behavior; the function value locates that point on the graph.
If the derivative is zero at one point, does that mean the function stops increasing there?
Not necessarily. f′(2) = 0 tells you the tangent slope is zero at that point. It does not, by itself, tell you what the function does on either side. Check the derivative’s sign in intervals to the left and right.
In the polynomial example, the derivative is negative before 2 and positive after 2, so the function changes from decreasing to increasing there. A zero derivative alone does not establish that change. The interval-wide sign pattern—not just the derivative’s value at the point—is what supports a conclusion about increasing or decreasing behavior [S1].
What should I write as my final conclusion, and how do I avoid overstating what the test shows?
Write the intervals and the reason for each conclusion. For the example, a clear answer is: “Since f′(x) = 2x − 4 is negative for x < 2 and positive for x > 2, f is decreasing on (−∞, 2) and increasing on (2, ∞).”
Keep the point separate from the intervals: f′(2) = 0, but that fact alone does not establish increasing or decreasing behavior on an interval. Also check the original function’s domain and where its derivative exists before claiming behavior across a boundary. The cited interval result requires continuity on the closed interval and differentiability in its interior [S1]. A test value is useful only when you have also established that the derivative keeps the same sign throughout that interval; one sample value alone does not establish that.
A practical checklist
Differentiate the function to find f′(x).
Mark points where f′(x) is zero or undefined, and respect the function’s domain.
Split the domain into intervals at those points.
Determine the derivative’s sign throughout each interval, using a test value when appropriate.
State increasing or decreasing for each interval, and treat boundary points separately.
A few more useful answers
Do I need to test more than one value in each interval?
A single test value is enough if you have established that the derivative cannot change sign within that interval. If its sign might change, split the interval further or justify the sign directly.
What if the derivative is undefined at a point?
Treat that point as a possible interval boundary and check the function’s domain and behavior on either side. An undefined derivative alone does not show whether the function is increasing or decreasing.
Does f′(x) > 0 at one point prove the function is increasing nearby?
Not on its own. It gives the slope at that point; an interval conclusion needs the derivative to have the appropriate sign throughout the interval, subject to the relevant continuity and differentiability conditions.
Sources
References used for this guide.
- [S1] The Mean Value Theorem — Calculus Volume 1, Section 4.4 openstax.org · Checked