Why Sign Changes Classify Local Extrema
A critical number is a place to investigate when identifying the important features of a graph. But being a candidate does not mean that the function has a local maximum or minimum there. The behavior on both sides matters: a function has a local maximum when it rises toward a point and then falls away, while it has a local minimum when it falls toward a point and then rises away.
The first derivative records whether a function is increasing or decreasing on an interval. Consequently, comparing the sign of the derivative just to the left and right of a critical number can reveal whether the function changes from rising to falling or from falling to rising. It can also show why a critical number where the derivative is zero need not be an extremum at all.
Local Maxima and Minima
Let \(f\) be defined on a domain containing \(c\). The point \(c\) is a local maximum number if there is an interval around \(c\) on which \(f(x)\leq f(c)\) for every \(x\) in the domain. It is a strict local maximum number if \(f(x)<f(c)\) for every such \(x\ne c\). Replace \(\leq\) by \(\geq\), and \(<\) by \(>\), to define a local minimum number and a strict local minimum number.
The corresponding point on the graph is \((c,f(c))\). These definitions are local: they compare \(f(c)\) with nearby function values, not with every value in the domain. A local maximum need not be the greatest value the function ever takes.
A critical number is a number \(c\) in the domain of \(f\) at which either \(f'(c)=0\) or \(f'(c)\) does not exist. Critical numbers are candidates for local extrema, but not every candidate is an extremum. The first-derivative sign test is useful when the derivative has a definite sign on each side of the candidate.
The First-Derivative Sign Change Test
Here, “just left” and “just right” mean throughout some interval on the corresponding side of \(c\). The theorem does not require \(f'(c)\) to exist. It does require \(f\) to be defined and continuous at \(c\), and the derivative signs to hold on the stated intervals. This makes the test applicable, for example, at a cusp where the derivative is undefined.
| Sign of \(f'\) left of \(c\) | Sign of \(f'\) right of \(c\) | Conclusion |
|---|---|---|
| Positive | Negative | Strict local maximum |
| Negative | Positive | Strict local minimum |
| Positive | Positive | Neither local maximum nor local minimum |
| Negative | Negative | Neither local maximum nor local minimum |
The last two rows require strict signs on both sides. If the derivative is zero throughout part of a neighboring interval, or its sign cannot be determined, this table alone may not settle whether there is a non-strict local extremum.
Why the Sign Change Test Works
The justification follows from the Mean Value Theorem. In the form needed here, it says that if a function \(f\) is continuous on a closed interval \([a,b]\) and differentiable on the open interval \((a,b)\), then there is some \(d\in(a,b)\) such that
Suppose \(f'(x)>0\) just to the left of \(c\). Take any nearby \(x<c\). The Mean Value Theorem applied on \([x,c]\) gives a number \(d\in(x,c)\) such that
Thus \(f(x)<f(c)\) for nearby \(x<c\): the function lies below its value at \(c\) on the left. If \(f'(x)<0\) just to the right, applying the theorem on \([c,x]\) gives \(f(x)-f(c)=f'(d)(x-c)<0\). The function also lies below \(f(c)\) on the right. Together, these inequalities show that \(c\) is a strict local maximum.
For the minimum case, the left-hand derivative is negative. For nearby \(x<c\), the same theorem gives \(f(c)-f(x)=f'(d)(c-x)<0\), so \(f(x)>f(c)\). On the right, a positive derivative gives \(f(x)-f(c)>0\), so again \(f(x)>f(c)\). Thus \(c\) is a strict local minimum.
If the derivative is positive on both sides, the same reasoning gives \(f(x)<f(c)\) to the left and \(f(x)>f(c)\) to the right. There are nearby values both below and above \(f(c)\), so \(c\) is neither type of local extremum. If the derivative is negative on both sides, the inequalities reverse, and there are again values both above and below \(f(c)\).
Worked Example: A Local Maximum and a Local Minimum
Worked Example: A Local Maximum and a Local Minimum
Classify the critical numbers of \(f(x)=x^3-3x\).
Differentiate and factor:
The derivative is zero when \(x=-1\) or \(x=1\). Check its sign on the three intervals determined by these critical numbers. At \(x=-2\), \(f'(-2)=3(4)-3=9>0\). At \(x=0\), \(f'(0)=-3<0\). At \(x=2\), \(f'(2)=12-3=9>0\). Therefore \(f'\) changes from positive to negative at \(-1\), and from negative to positive at \(1\).
The sign test classifies \(-1\) as a strict local maximum number and \(1\) as a strict local minimum number. Evaluate the function to find the graph points:
The local maximum point is \((-1,2)\), and the local minimum point is \((1,-2)\). As a check, the derivative signs show the function rises before \(-1\) and falls after it, then falls before \(1\) and rises after it. This matches the classifications.
Worked Example: A Zero Derivative Without an Extremum
Worked Example: A Zero Derivative Without an Extremum
Determine whether \(f(x)=x^3\) has a local extremum at \(x=0\).
Differentiate:
At \(x=0\), \(f'(0)=3(0)^2=0\), so \(0\) is a critical number. For every \(x<0\) and every \(x>0\), however, \(3x^2>0\). The derivative is positive on both sides: its sign does not change at \(0\).
By the sign test, \(0\) is neither a local maximum number nor a local minimum number. Direct comparison confirms this. For any nearby \(x<0\), \(x^3<0=f(0)\); for any nearby \(x>0\), \(x^3>0=f(0)\). Thus nearby values occur both below and above \(f(0)\), as required for neither type of extremum.
The conclusion is not that the function stops changing at \(0\). It continues to increase through \(0\). The derivative being zero at one point does not, by itself, imply a turning point.
Worked Example: A Minimum Where the Derivative Is Undefined
Worked Example: A Minimum Where the Derivative Is Undefined
Classify the critical number \(x=0\) for \(f(x)=|x|\).
The function is continuous at \(0\), and \(f(0)=|0|=0\). For \(x<0\), \(|x|=-x\), so \(f'(x)=-1<0\). For \(x>0\), \(|x|=x\), so \(f'(x)=1>0\). The derivative is undefined at \(0\), but its sign changes from negative on the left to positive on the right.
The sign test therefore gives a strict local minimum at \(0\). In fact, for every \(x\ne0\), \(|x|>0=|0|\), which directly verifies the strict minimum. The graph point is \((0,0)\).
Reading Sign Changes Carefully
When using this test, the signs must be determined on intervals, not inferred from a single value at the critical number. If \(f'(c)=0\), that tells you what the derivative is at \(c\); it does not tell you whether the derivative is positive or negative nearby. To classify the point, examine the signs on both sides.
A sign chart is one way to organize that examination. Locate the critical numbers, divide the domain into intervals at those numbers, and determine the sign of \(f'\) on each interval. Then compare the signs that meet at each candidate. The conclusion concerns the behavior near the candidate, so only the neighboring intervals are needed for that classification.
The test also has limits. If the derivative has the same strict sign on both sides, the candidate is not a local extremum. But if the derivative is zero on a neighboring interval, or the sign is not constant on either side, the basic sign comparison may be inconclusive; further reasoning from the function’s values may be needed. A conclusion from the test always depends on its conditions, including continuity at \(c\).
Local Behavior Is Not Endpoint or Global Behavior
The sign-change test describes a two-sided neighborhood of a point. It does not automatically classify an endpoint of the domain, where there may be no domain points on one side. Endpoints can be local extrema under definitions that compare only with nearby points in the domain, but their classification requires a one-sided comparison instead of the two-sided test stated above.
Nor does a local classification establish that a point is a global maximum or minimum. A strict local maximum is higher than nearby values; a global maximum must be at least as high as every function value in the entire domain. To determine global behavior, the domain and the function’s values beyond the immediate neighborhood must also be considered.
For curve analysis, the first-derivative sign change gives a concise description of a turning point: positive to negative means rising then falling; negative to positive means falling then rising. The cases with no change are equally informative. They prevent a zero derivative or an undefined derivative from being mistaken for an extremum when the function does not actually turn.
Check Your Understanding
Use the first-derivative sign test where its conditions apply.
- If \(f'\) changes from negative to positive at a critical number \(c\), what type of local extremum occurs?
- For \(f(x)=x^3\), why does \(f'(0)=0\) not imply a local maximum or minimum?
- What continuity condition is included in the stated sign-change theorem?
- If \(f'\) is negative on both sides of a critical number, does the sign test give a local extremum?
- For \(f(x)=|x|\), what derivative signs on either side of zero lead to the classification at zero?
