When a Derivative Test Does Not Decide
A derivative test is a way to use information about a function’s derivatives to classify a point as a local maximum, a local minimum, or neither. But a test gives a conclusion only when its conditions are met and its result is decisive. A derivative value at one point is not, by itself, a complete description of how the function behaves nearby.
This distinction matters in two situations. A test may be inapplicable because a required derivative or domain condition fails. Or it may be applicable but inconclusive, meaning the result does not tell us which classification is correct. In either situation, the right response is not to guess: look for additional information, often the signs of the first derivative immediately to either side of the point.
Local Extrema and the Scope of the Tests
A function \(f\) has a local maximum at \(x=c\) if there is an interval around \(c\) such that \(f(c)\ge f(x)\) for every \(x\) in the domain within that interval. A local minimum is defined by reversing the inequality: \(f(c)\le f(x)\). If the relevant inequalities are strict whenever \(x\ne c\), the extremum is strict.
The familiar derivative tests are principally about points in the interior of the domain. If \(f\) is differentiable at an interior local extremum \(c\), Fermat’s theorem says that \(f'(c)=0\). A point where \(f'(c)=0\), or where \(f'(c)\) does not exist, is called a critical point when \(c\) is in the domain. Critical points are candidates to investigate, not automatic extrema.
The first derivative test uses the sign of \(f'\) on intervals to the left and right of \(c\). In its strict-sign form, it requires \(f\) to be continuous at \(c\), differentiable on the nearby intervals on each side, and the derivative to have a consistent, nonzero sign on each side. The second derivative test requires more: in the version used here, \(f'(c)=0\), and \(f''\) exists and is continuous in a neighborhood of \(c\). If \(f''(c)>0\), the test gives a local minimum; if \(f''(c)<0\), it gives a local maximum. If \(f''(c)=0\), this test is inconclusive.
These are sufficient tests: when their conditions and decisive sign conclusions hold, they establish an extremum. If a condition fails, or a result such as \(f''(c)=0\) gives no direction, the classification must be determined another way.
Why Nearby First-Derivative Signs Decide the Case
The first derivative describes whether the function is increasing or decreasing. If the function increases as it approaches \(c\) from the left and decreases just after \(c\), its nearby values rise toward \(f(c)\) and then fall away. That is the pattern of a local maximum. The reverse pattern produces a local minimum.
If \(f'(x)>0\) immediately to the left of \(c\), then \(f\) is increasing there; if \(f'(x)<0\), it is decreasing.
Determine whether \(f\) increases or decreases immediately to the right of \(c\).
A change from positive to negative gives a local maximum; a change from negative to positive gives a local minimum. No change between strict signs gives neither.
Here is a justification for the positive-to-negative case. Suppose \(f\) is continuous at \(c\), differentiable on each side nearby, \(f'(x)>0\) just to the left, and \(f'(x)<0\) just to the right. For any nearby \(x<c\), the Mean Value Theorem applied on \([x,c]\) gives some \(\xi\in(x,c)\) such that
Thus \(f(c)>f(x)\) on the left. For any nearby \(x>c\), the Mean Value Theorem on \([c,x]\) gives some \(\eta\in(c,x)\) such that
Thus \(f(c)>f(x)\) on the right as well, proving that \(c\) is a strict local maximum. Reversing the two derivative signs proves the strict local minimum case. If the derivative has the same strict sign on both sides, the function keeps moving in the same direction through \(c\), so there is no local extremum there.
Worked Example: A Zero Second Derivative but No Extremum
Worked Example: A Zero Second Derivative but No Extremum
Classify \(c=0\) for \(f(x)=x^3\), and explain what the derivative tests establish.
First evaluate the function at the point: \(f(0)=0^3=0\). Differentiate:
Differentiate again:
The second derivative test is inconclusive because \(f''(0)=0\). Now check the first-derivative signs. For every \(x\ne0\), \(3x^2>0\), so \(f'(x)>0\) on both sides of zero. The derivative does not change sign, and the function is increasing through the point.
Direct function values confirm the classification: if \(x<0\), then \(x^3<0=f(0)\); if \(x>0\), then \(x^3>0=f(0)\). Nearby values occur both below and above \(f(0)\), so zero is neither a local maximum nor a local minimum. The first-derivative signs and the function values agree.
Worked Example: The Same Second-Derivative Result, Different Outcomes
Worked Example: The Same Second-Derivative Result, Different Outcomes
Compare \(f(x)=x^4\) and \(g(x)=-x^4\) at \(x=0\). Both have a zero second derivative there, but their classifications differ.
For \(f(x)=x^4\),
The second derivative test is inconclusive. For \(x<0\), \(4x^3<0\); for \(x>0\), \(4x^3>0\). Thus \(f'\) changes from negative to positive, which gives a local minimum. Directly, \(x^4>0=f(0)\) for every \(x\ne0\), confirming a strict local minimum.
For \(g(x)=-x^4\),
Again, the second derivative test is inconclusive. For \(x<0\), \(-4x^3>0\); for \(x>0\), \(-4x^3<0\). The first derivative changes from positive to negative, giving a local maximum. Directly, \(-x^4<0=g(0)\) for every \(x\ne0\), confirming a strict local maximum. In each case, the nearby first-derivative signs supply the information that \(f''(0)\) does not.
When a Test Cannot Be Applied
A test may fail to apply even when an extremum is present. For example, a corner can prevent the first derivative from existing at the point, so Fermat’s theorem and the second derivative test do not apply there. But the first-derivative sign pattern on either side can still be useful. Likewise, a point at the boundary of a domain is not an interior point, so Fermat’s theorem does not require its derivative to be zero.
Worked Example: A Corner Classified by One-Sided Derivative Signs
Classify \(x=0\) for \(f(x)=|x|\).
The function value is \(f(0)=|0|=0\). For \(x<0\), \(|x|=-x\), so \(f'(x)=-1\). For \(x>0\), \(|x|=x\), so \(f'(x)=1\). The derivative changes from negative to positive across zero, indicating a local minimum.
The derivative at zero does not exist: the left-hand derivative is \(-1\), while the right-hand derivative is \(1\). Therefore, tests requiring \(f'(0)=0\) cannot be used. But direct comparison verifies the conclusion: \(|x|>0=f(0)\) for every \(x\ne0\). Thus zero is a strict local minimum. The one-sided signs help even though the derivative is undefined at the point.
Worked Example: An Endpoint Outside Fermat’s Theorem
Let \(f(x)=x\) on the domain \([0,2]\). Determine whether \(x=0\) is a local extremum relative to this domain.
For every \(x\in[0,2]\), \(f(x)=x\ge0=f(0)\), with strict inequality when \(x>0\). Therefore, zero is a strict local minimum relative to the domain (and also an absolute minimum on this domain).
The formula has derivative \(f'(x)=1\), including a right-hand derivative at zero. That derivative is not zero. There is no contradiction with Fermat’s theorem: zero is a domain endpoint, not an interior point, so the theorem’s condition is unmet. The comparison with domain values verifies the result.
A Common Mistake: Treating “Inconclusive” as “Neither”
A frequent error is to treat a test that gives no answer as though it had proved that no extremum exists. For instance, someone might calculate \(f''(0)=0\) for \(f(x)=x^4\) and conclude, “There is no local extremum at zero.” That conclusion does not follow. The second derivative test is inconclusive in this case, while the first derivative changes from negative to positive and the function values satisfy \(x^4>0=f(0)\) for \(x\ne0\). Zero is a strict local minimum.
The correction is to separate the test’s output from the classification: “The second derivative test does not decide this point. I will check the first-derivative signs or compare function values nearby.” This is especially important when a derivative does not exist or when the point lies at a domain boundary. Those facts limit which tests can be used; they do not settle the extremum question by themselves.
What Additional Information Is Enough?
When a derivative test does not decide, the needed information depends on what failed. For an interior point where the second derivative test gives zero, a first-derivative sign chart often settles the classification. If the first derivative is unavailable at the point, its signs on either side may still decide a corner. At an endpoint, compare the function with values from the domain side. In every case, conclusions must use a neighborhood or the domain values relevant to the definition of a local extremum.
| Evidence near \(c\) | Conclusion |
|---|---|
| \(f'\) changes from positive to negative | Strict local maximum |
| \(f'\) changes from negative to positive | Strict local minimum |
| \(f'\) has the same strict sign on both sides | Neither a local maximum nor a local minimum |
| \(f''(c)=0\) in the second derivative test | No conclusion from that test; obtain additional evidence |
| A derivative condition fails at a corner or endpoint | That test does not apply; inspect one-sided behavior or compare values |
The strict sign patterns in the table are clear, sufficient conditions. If the derivative is zero at other nearby points or does not keep a consistent sign, a simple sign-change conclusion may not be available. Then use stronger information, such as direct comparisons with \(f(c)\), rather than extending the test beyond its conditions.
Check Your Understanding
For each question, distinguish what a derivative test establishes from what still needs to be checked.
- If \(f'(c)=0\) and \(f''(c)=0\), what can be concluded from the second derivative test alone?
- Suppose \(f'\) is negative just to the left of \(c\) and positive just to the right. What classification follows, assuming the first derivative test’s conditions hold?
- Why does a local extremum at an endpoint not contradict Fermat’s theorem?
- For \(f(x)=x^3\) at zero, what do the first- and second-derivative tests show, and what do function values confirm?
- At a corner where \(f'(c)\) does not exist, what nearby information may still classify the point?
