First Derivative Test
In short: First derivative test decides whether a flat or broken slope marks a peak, a valley or neither, by checking how the sign of f’ behaves just before and just after it.
AP Calculus ABCalculusAP Calculus AB Unit 5
What is First Derivative Test?
The first derivative test reads a graph’s story from the sign of its derivative. Positive means the function is climbing, negative means it is falling, so a switch of sign at a critical point reveals a turning point.
Take a critical number \(c\) where \(f\) is continuous. Positive just before \(c\) and negative just after: \(f(c)\) is a local maximum. Negative before and positive after: \(f(c)\) is a local minimum. Same sign on both sides: nothing special happens at \(c\).
How to spot it
- The problem wants the stretches where a graph climbs or drops and the spots where it turns around.
- You are handed a graph or table of f’ and must decide what f is doing.
- A sign chart or number line with test values is expected in the justification.
- The second derivative is messy, zero, or not allowed, so signs of f’ are the safer tool.
The method
- Compute \(f^{\prime}(x)\) and list the critical numbers by solving \(f^{\prime}(x) = 0\) or finding where it is undefined.
- Mark those numbers on a number line; they split it into intervals.
- Pick one test value in each interval and record the sign of \(f^{\prime}\).
- Apply the first derivative test at each critical number and report the extreme value \(f(c)\).
First Derivative Test: solved example
Problem
Classify the critical points of \(f(x) = x^3 – 3x^2 – 9x + 4\).
Solution
\(f^{\prime}(x) = 3x^2 – 6x – 9 = 3(x – 3)(x + 1)\), so the critical numbers are \(-1\) and \(3\).
Test values: \(f^{\prime}(-2) = 15 > 0\); \(f^{\prime}(0) = -9 < 0\); \(f^{\prime}(4) = 15 > 0\).
The sign goes from \(+\) to \(-\) at \(-1\): local maximum \(f(-1) = -1 – 3 + 9 + 4 = 9\). It goes from \(-\) to \(+\) at 3: local minimum \(f(3) = 27 – 27 – 27 + 4 = -23\).
Try it
Question 1
For \(g(x) = x^3\), the derivative \(3x^2\) is zero at \(x = 0\). Is there a local extreme there?
Show the answer
No. \(g^{\prime}\) is positive on both sides of 0, so the sign does not change and 0 is neither a maximum nor a minimum.
Question 2
If \(f^{\prime}(x) = (x – 5)(x + 2)^2\), what happens at \(x = 5\) and \(x = -2\)?
Show the answer
At 5 the sign changes from negative to positive: a local minimum. At \(-2\) the squared factor never changes sign, so there is no extreme.
First Derivative Test vs. Second Derivative Test
The first derivative test looks at signs of \(f^{\prime}\) on both sides of a critical number and works even when the second derivative is zero. The second derivative test looks only at the sign of \(f^{\prime\prime}\) at the point itself and can be inconclusive. Neither replaces the other in every situation.
Keep going
Lessons
Worksheets and quizzes
- Increasing, Decreasing and the First Derivative Test – Worksheet
- Increasing, Decreasing and the First Derivative Test – Quiz
Related words
Whole dictionary: A–Z · AP Calculus AB · Calculus