Why These Categories Matter
A search for local maxima and minima begins by separating points into useful categories. At an interior point, an extremum can occur where the derivative is zero or where the derivative does not exist. But an extremum can also occur at an endpoint, where the usual two-sided derivative test does not apply.
These categories overlap in purpose: each identifies places worth checking. They do not have the same definition, and none of them guarantees that an extremum is present. A stationary point, for example, can lie on a graph that simply passes through with a horizontal tangent.
Stationary Points
The derivative describes the slope of the tangent line when the derivative exists. Thus, at a stationary point, the tangent line is horizontal. This is a statement about the local slope, not about whether nearby function values are larger or smaller.
Worked Example: A Stationary Point That Is a Local Minimum
Let \(f(x)=x^2\), with domain \(\mathbb{R}\). First find the derivative:
Set the derivative equal to zero:
Therefore, \(x=0\) is a stationary number, and the stationary point is \((0,f(0))=(0,0)\). To check its extremum status, compare nearby values with \(f(0)\): for every \(x\ne 0\), \(x^2>0=f(0)\). Hence \(x=0\) is a strict local minimum (and also a global minimum). The derivative confirms stationarity; the comparison of function values confirms the minimum.
Check: \(f'(0)=2(0)=0\), as required for a stationary number.
Critical Numbers and Differentiability
This convention keeps endpoints separate from interior critical numbers. Some textbooks use “critical number” more broadly and may include domain endpoints. Here, endpoints are treated as their own category, so the interior condition in the definition matters.
A nondifferentiable critical number is not a stationary number: stationarity requires that the derivative exist and equal zero. A sharp corner, a vertical tangent, or another failure of differentiability may produce a nondifferentiable critical number. Such a point may be an extremum, but it need not be.
Worked Example: A Nondifferentiable Critical Number That Is a Local Minimum
Consider \(f(x)=|x|\), with domain \(\mathbb{R}\). For \(x<0\), \(f(x)=-x\), so \(f'(x)=-1\). For \(x>0\), \(f(x)=x\), so \(f'(x)=1\). At \(x=0\), the left-hand derivative is \(-1\) and the right-hand derivative is \(1\), so the derivative does not exist there.
Thus \(0\) is an interior critical number, specifically a nondifferentiable critical number. It is not stationary. To determine whether it is an extremum, note that \(|x|\ge 0=|0|\) for every \(x\), with strict inequality when \(x\ne 0\). Therefore, \(x=0\) is a strict local minimum (and a global minimum).
Check: the two one-sided derivatives disagree, confirming nondifferentiability; the function-value comparison confirms the minimum.
Worked Example: A Nondifferentiable Critical Number That Is Not an Extremum
Let \(g(x)=\sqrt[3]{x}\), with domain \(\mathbb{R}\). For \(x\ne 0\),
As \(x\) approaches \(0\), this derivative is unbounded, so \(g'(0)\) does not exist as a finite derivative. The number \(0\) is therefore a nondifferentiable critical number. However, \(g\) is strictly increasing: if \(x_1<x_2\), then \(\sqrt[3]{x_1}<\sqrt[3]{x_2}\). In particular, values just below \(0\) are negative and values just above \(0\) are positive. Neither all nearby values are at least \(g(0)=0\), nor are they all at most \(g(0)\). Hence \(0\) is neither a local minimum nor a local maximum.
Check: \(g(0)=0\); for every \(h>0\), \(g(-h)<g(0)<g(h)\), ruling out both types of local extremum.
Endpoints Are Separate Candidates
For a function defined on an interval, an endpoint is a boundary value of that interval that belongs to the domain. For example, \(0\) and \(2\) are endpoints of \([0,2]\). The endpoints of \((0,2)\) are not in the domain, so they cannot themselves be local extrema of a function whose domain is only \((0,2)\).
Local extrema at an endpoint are understood relative to the domain: only nearby points that belong to the domain are considered. Thus, at the left endpoint of \([a,b]\), comparisons are made with nearby points to its right; at the right endpoint, with nearby points to its left. This one-sided comparison can establish an endpoint extremum even though the ordinary two-sided derivative is not defined there.
Worked Example: Endpoints That Are Local Extrema
Define \(h(x)=(x-1)^2\) on the closed interval \([0,2]\). At the left endpoint, \(h(0)=1\). For \(0<x<1\), we have \(0<1-x<1\), so \(0<(1-x)^2<1\). Thus, for domain points sufficiently close to \(0\), the function values are less than \(h(0)\); \(0\) is a local maximum relative to the domain.
At the right endpoint, \(h(2)=1\). For \(1<x<2\), \(0<x-1<1\), so \(0<(x-1)^2<1\). Therefore, \(2\) is also a local maximum relative to the domain. At the interior point \(x=1\), \(h(1)=0\), and \(h(x)\ge 0\) everywhere, so \(1\) is a local minimum.
Check: the endpoint values are \(h(0)=1\) and \(h(2)=1\); nearby interior values are less than \(1\), confirming both endpoint maxima.
Worked Example: An Endpoint That Is Not a Local Extremum
Define \(q\) on \([0,1]\) by \(q(0)=0\) and \(q(x)=x\sin(1/x)\) for \(0<x\le 1\). The point \(0\) is an endpoint in the domain. To test whether it is a local extremum, consider the positive sequences
Both sequences consist of positive numbers in the domain and tend to \(0\). Also, \(\sin(1/a_n)=1\), while \(\sin(1/b_n)=-1\). Consequently,
There are domain points arbitrarily close to \(0\) where the function is above \(q(0)\), and others where it is below \(q(0)\). Therefore \(0\) is neither a local maximum nor a local minimum. The endpoint status identifies a point to check; it does not decide the result.
Check: \(a_n\to0\) and \(b_n\to0\), while their function values have opposite signs, so neither one-sided extremum condition holds.
Why These Points Are Candidates
Fermat’s theorem gives the key restriction: if a function has a local extremum at an interior point and is differentiable there, then its derivative at that point is zero. This does not say every zero of the derivative is an extremum. It says an interior extremum with a derivative cannot occur at a point where the derivative is nonzero.
The candidate list follows by separating the possibilities for an interior local extremum. Either the derivative exists or it does not. If it exists, Fermat’s theorem requires it to be zero, making the point stationary. If it does not exist, the point is a nondifferentiable critical number. A point at the boundary is not covered by this interior argument and must be considered separately.
These are stationary numbers. Test the function values around each point; stationarity alone does not establish an extremum.
These are nondifferentiable critical numbers. They may be extrema, as with \(|x|\) at \(0\), or not, as with \(\sqrt[3]{x}\) at \(0\).
Use nearby domain points on the available side or sides. Do not apply the interior conclusion of Fermat’s theorem at an endpoint.
This reasoning is a necessary-condition argument, not a method that automatically classifies each candidate. To decide whether a candidate is a local maximum, a local minimum, or neither, compare function values in the relevant neighborhood. In particular, a horizontal tangent or a missing derivative is evidence to investigate, not a verdict.
Common Pitfall: Treating Every Candidate as an Extremum
A frequent error is to find a zero derivative and immediately announce a local maximum or minimum. For \(f(x)=x^3\), the derivative is \(f'(x)=3x^2\), so \(f'(0)=0\). Yet if \(x<0\), then \(x^3<0\), and if \(x>0\), then \(x^3>0\). Values on both sides of \(f(0)=0\) occur arbitrarily close to \(0\), so \(0\) is not a local extremum.
The incorrect conclusion “\(f'(0)=0\), therefore \(0\) is a local minimum” confuses a necessary condition with a sufficient one. The correction is to use the derivative equation only to identify a candidate, then check nearby function values or use an appropriate classification method.
Likewise, a nondifferentiable critical number need not be an extremum, and an endpoint need not be one either. The three categories organize the search; the local-extremum definition determines the answer.
Summary
| Category | Meaning | Does it guarantee a local extremum? |
|---|---|---|
| Stationary number | An interior point where \(f'(c)=0\) | No |
| Nondifferentiable critical number | An interior point where \(f'(c)\) does not exist | No |
| Included endpoint | A boundary point that belongs to the domain | No |
For an interior local extremum, Fermat’s theorem leaves two possibilities: the derivative is zero, or the derivative does not exist. Endpoints are checked separately because the interior theorem does not apply there. After identifying these candidates, compare function values to determine whether an extremum actually occurs.
Check Your Understanding
For each question, distinguish identifying a candidate from proving that it is an extremum.
- What condition makes an interior point a stationary number?
- Can a nondifferentiable critical number be a local extremum? Give an example from this tutorial.
- Why is \(x=0\) for \(f(x)=x^3\) a stationary number but not a local extremum?
- How should local extrema at an endpoint of \([a,b]\) be tested?
- Does being an endpoint guarantee a local maximum or minimum? Explain briefly.
