Introduction
A critical number identifies a point worth examining, but it does not by itself tell us whether the function has a local maximum, a local minimum, or neither there. The First Derivative Test answers that question by comparing the signs of the derivative immediately to the left and right of the critical number.
The test turns a first-derivative sign chart into a local classification. Its conclusions are local: they describe the function near the critical number, not necessarily its behavior over its entire domain.
What the Test Requires
Let \(c\) be an interior point of the domain of \(f\). Recall that \(c\) is a critical number if \(f'(c)=0\) or \(f'(c)\) does not exist. For this test, we also need \(f\) to be continuous at \(c\), and the derivative to have a definite, strict sign on each side of \(c\) in some interval around it. The derivative need not exist at \(c\).
“On each side” means throughout a sufficiently small open interval to the left and a sufficiently small open interval to the right—not merely at one sample point on each side. The intervals must lie in the domain, and the derivative must exist at their points.
The sign patterns have the following meanings:
| Sign of \(f'\) to the left | Sign of \(f'\) to the right | Classification at \(c\) |
|---|---|---|
| Positive | Negative | Local maximum |
| Negative | Positive | Local minimum |
| Positive | Positive | Neither |
| Negative | Negative | Neither |
A positive derivative means the function is increasing on the interval in question; a negative derivative means it is decreasing there. This connection between derivative signs and monotonicity is the key to reading the table.
Why the Sign Pattern Gives the Classification
We can justify the test using the monotonicity result: if a function is continuous on an interval and differentiable in its interior, then a strictly positive derivative throughout the interior makes the function strictly increasing, while a strictly negative derivative makes it strictly decreasing.
Suppose first that \(f'(x)>0\) just to the left of \(c\) and \(f'(x)<0\) just to its right. The function increases toward \(c\) from the left and decreases away from \(c\) to the right. Continuity at \(c\) joins those behaviors at its function value. To see that the comparison is strict, take any nearby \(x<c\) and choose \(y\) with \(x<y<c\). Strict increase gives \(f(x)<f(y)\), and continuity at \(c\), together with increase up to \(c\), gives \(f(y)\le f(c)\). Thus \(f(x)<f(c)\). The corresponding argument on the right gives \(f(x)<f(c)\) for nearby \(x>c\). Hence \(c\) is a strict local maximum.
For a negative-to-positive change, the function decreases toward \(c\) and increases away from it. The same monotonicity and continuity reasoning gives \(f(x)>f(c)\) for nearby \(x\ne c\), so \(c\) is a strict local minimum. If the derivative has the same strict sign on both sides, the function moves in the same direction on both sides of \(c\); there are nearby function values both below and above \(f(c)\). Therefore \(c\) is neither a local maximum nor a local minimum.
The test concerns signs, not the size of the derivative. For instance, a derivative can approach zero near \(c\) and still remain positive throughout the interval to the left. Its sign, rather than its magnitude, determines whether the function is increasing or decreasing there.
Worked Example: A Positive-to-Negative Change
Worked Example: Classifying a Local Maximum
Let \(f(x)=-x^2+4x+1\). Use the First Derivative Test to classify the critical number.
Differentiate:
Set the derivative equal to zero:
The function is a polynomial, so it is continuous and differentiable everywhere. In particular, it is continuous at \(c=2\). Check the derivative sign on each side:
Thus \(f'\) changes from positive to negative. The test classifies \(x=2\) as a strict local maximum. Its function value is
Check: At \(x=1\), \(f(1)=-1+4+1=4\), which is below \(5\); at \(x=3\), \(f(3)=-9+12+1=4\), also below \(5\). These values agree with the predicted local maximum. The derivative signs on the intervals, rather than these two checks alone, establish the classification.
Worked Example: A Negative-to-Positive Change
Worked Example: Classifying a Local Minimum at a Cusp
Let \(f(x)=|x|\). Classify the critical number at \(x=0\).
The function is continuous at \(0\), with \(f(0)=0\). For \(x<0\), \(f(x)=-x\), so \(f'(x)=-1\). For \(x>0\), \(f(x)=x\), so \(f'(x)=1\). The derivative at \(0\) does not exist because the left and right derivatives differ. Thus \(0\) is a critical number, and the derivative signs are
The sign changes from negative to positive, so the First Derivative Test classifies \(0\) as a strict local minimum.
Check: If \(x\ne0\), then \(|x|>0=|0|\). Hence nearby function values are strictly greater than \(f(0)\), as the test predicts. This example also shows that the test applies when the derivative is undefined at the critical number, provided the function is continuous there and the side conditions hold.
Worked Example: No Local Extremum Despite a Critical Number
Worked Example: Same Sign on Both Sides
Let \(f(x)=x^3\). Classify its critical number at \(x=0\).
Differentiate:
The derivative is zero at \(x=0\), so \(0\) is a critical number. For every \(x\ne0\), \(3x^2>0\). In particular, \(f'\) is positive on an interval to the left of \(0\) and on an interval to the right of \(0\). Its sign does not change. By the test, \(0\) is neither a local maximum nor a local minimum.
Check: For any \(x<0\), \(x^3<0=f(0)\); for any \(x>0\), \(x^3>0=f(0)\). Thus arbitrarily close to \(0\), there are function values below and above \(f(0)\). This confirms that \(0\) is neither type of local extremum.
When the Test Does Not Decide
The First Derivative Test gives a classification only when the derivative has a strict, constant sign on each side of \(c\) and the function is continuous at \(c\). If the derivative is zero at some points on those side intervals, changes sign more than once, or has no fixed sign there, the simple four-row table does not apply. Likewise, a derivative sign pattern alone cannot classify a point if continuity at \(c\) is not known.
An inconclusive application is not a claim that the point is neither type of extremum. It means only that this test has not settled the question. Other information about the function may be needed.
Common Mistake: Reversing the Sign Pattern
A frequent error is to associate positive-to-negative with a minimum. That reverses the direction of the function’s movement. When \(f'>0\), the function increases; when \(f'<0\), it decreases. An increase followed by a decrease produces a peak, not a trough.
For example, suppose a sign chart shows \(f'>0\) to the left of \(c\) and \(f'<0\) to the right. The incorrect conclusion would be “local minimum.” The correction is to read the motion in order: the function rises as \(x\) approaches \(c\), then falls as \(x\) moves past \(c\). Therefore \(c\) is a local maximum.
A second error is to stop as soon as \(f'(c)=0\). In the example \(f(x)=x^3\), the derivative is zero at \(0\), but the derivative remains positive on both sides and the function has no local extremum there. Always inspect both sides.
Using the Test Reliably
For a given critical number, organize the reasoning in a fixed order:
Check that \(c\) is an interior point of the domain and that \(f\) is continuous at \(c\).
Use intervals around \(c\), and establish a strict sign throughout each interval rather than relying on isolated test values.
Positive-to-negative means local maximum; negative-to-positive means local minimum; the same strict sign on both sides means neither.
The result is local. It does not, by itself, establish an absolute maximum or minimum over a larger domain.
The First Derivative Test is useful because it classifies a critical number from local behavior, including at points where the derivative is undefined. A correctly constructed sign chart provides the needed evidence, while the continuity and interval conditions ensure that the conclusion applies to the function itself.
Check Your Understanding
For each question, assume the function is continuous at the critical number and the stated derivative signs hold throughout intervals immediately to either side.
- If \(f'\) is negative to the left of \(c\) and positive to the right, what does the test conclude?
- If \(f'\) is positive on both sides of \(c\), is \(c\) a local maximum, a local minimum, or neither?
- Does the test require \(f'(c)=0\), or can it apply when \(f'(c)\) is undefined?
- What does it mean if \(f'\) does not have a fixed strict sign on one of the side intervals?
- Why is knowing only that \(f'(c)=0\) insufficient to classify \(c\)?
