From Critical Numbers to a Sign Chart
A derivative sign chart organizes information about a function’s behavior. It shows the critical numbers that divide the domain into intervals, the sign of the derivative on each interval, and whether the function is increasing or decreasing there. The chart is a record of derivative signs and monotonicity; it does not, by itself, require us to classify local extrema.
The construction depends on two earlier ideas: critical numbers identify important points in the domain, and a function is increasing or decreasing on an interval when its derivative has the corresponding sign throughout that interval. The practical task is to organize those ideas without missing a point or testing the derivative at a point where it is zero or undefined.
Preparing the Intervals
Begin with the domain of \(f\), not just the formula for \(f'\). A critical number is a number \(c\) in the domain of \(f\) at which \(f'(c)=0\) or \(f'(c)\) does not exist. A point where the function itself is not defined is not a critical number, even if the derivative formula has a denominator that vanishes there.
After identifying the critical numbers, place them in increasing order. They divide an interval domain into open intervals. These intervals are where you test the derivative: the critical numbers themselves are boundaries between intervals, not test points. If a domain has an endpoint, there may be only one interval on the endpoint’s side that lies in the domain.
On each interval, determine whether \(f'\) is positive or negative. You can evaluate \(f'\) at a convenient test point, or determine its sign from its factors. For the test-point method, the derivative must be defined throughout the interval. If \(f'\) is continuous there and has no zero there, its sign cannot change inside the interval: a sign change would force it to take the value zero by the Intermediate Value Theorem. The derivatives of the elementary functions in the examples below have this continuity property on the intervals being tested.
A Reliable Construction
Record where \(f\) is defined. Do not treat a point outside the domain as a critical number.
Within the domain, solve \(f'(x)=0\) and find points where \(f'\) does not exist.
List the critical numbers from least to greatest and write the open intervals they determine in the domain.
Use a test point in each interval or analyze the signs of the derivative’s factors.
Write “increasing” for a positive derivative and “decreasing” for a negative derivative on each interval.
Worked Examples
Worked Example: A Cubic with Two Critical Numbers
Construct a first-derivative sign chart for \(f(x)=x^3-3x\).
Step 1: Find the derivative and critical numbers. The function is a polynomial, so its domain is \(\mathbb{R}\). Differentiating gives
Setting \(f'(x)=0\) gives \(3(x-1)(x+1)=0\), so the critical numbers are \(x=-1\) and \(x=1\). The derivative exists everywhere, so there are no other critical numbers.
Step 2: Form the intervals and test each one. In order, the critical numbers divide the real line into \((-\infty,-1)\), \((-1,1)\), and \((1,\infty)\). Choose the test points \(-2\), \(0\), and \(2\), respectively:
There are no zeros of \(f'\) inside these intervals, and \(f'\) is continuous, so each test gives the sign throughout its interval.
| Interval | Test point and derivative value | Sign of \(f'\) | Behavior of \(f\) |
|---|---|---|---|
| \((-\infty,-1)\) | \(f'(-2)=9\) | Positive | Increasing |
| \((-1,1)\) | \(f'(0)=-3\) | Negative | Decreasing |
| \((1,\infty)\) | \(f'(2)=9\) | Positive | Increasing |
Check: Substituting the two critical numbers into the factored derivative confirms \(f'(-1)=3(-2)(0)=0\) and \(f'(1)=3(0)(2)=0\). The test points lie in their stated intervals, and the computed signs agree with the chart.
Worked Example: A Quartic with Three Critical Numbers
Construct a sign chart for \(g(x)=x^4-4x^2\).
Step 1: Find and factor the derivative. The domain is \(\mathbb{R}\), and
Thus \(g'(x)=0\) at \(x=-\sqrt{2}\), \(x=0\), and \(x=\sqrt{2}\). The derivative exists at every real number. In increasing order, the critical numbers divide the real line into four intervals.
Step 2: Test each interval. Use \(-2\), \(-1\), \(1\), and \(2\) as test points. The factored derivative gives:
Since \(g'\) is continuous and has no zeros within any of the four intervals, these signs hold throughout their respective intervals.
| Interval | Test point and derivative value | Sign of \(g'\) | Behavior of \(g\) |
|---|---|---|---|
| \((-\infty,-\sqrt{2})\) | \(g'(-2)=-16\) | Negative | Decreasing |
| \((-\sqrt{2},0)\) | \(g'(-1)=4\) | Positive | Increasing |
| \((0,\sqrt{2})\) | \(g'(1)=-4\) | Negative | Decreasing |
| \((\sqrt{2},\infty)\) | \(g'(2)=16\) | Positive | Increasing |
Check: The factored expression gives \(g'(-\sqrt{2})=0\), \(g'(0)=0\), and \(g'(\sqrt{2})=0\). Each test point belongs to the interval where it is used. The chart lists all three zeros and gives a derivative sign for each intervening interval.
Worked Example: A Domain Endpoint Where the Derivative Does Not Exist
Construct a sign chart for \(h(x)=\sqrt{x}\), whose domain is \([0,\infty)\).
For \(x>0\), differentiation gives
This derivative is positive for every \(x>0\). At \(x=0\), the function is defined, but the derivative does not exist as a finite real number: for \(t>0\), the difference quotient is
which grows without bound as \(t\) approaches \(0\) from the right. Thus \(0\) is a critical number because it belongs to the domain and \(h'(0)\) does not exist. It is also the domain’s left endpoint, so there is no interval to its left in the domain.
Choose \(x=1\) to test the interval \((0,\infty)\):
| Domain point or interval | Information | Behavior of \(h\) |
|---|---|---|
| \(0\) | In the domain; \(h'(0)\) does not exist | Critical number and endpoint |
| \((0,\infty)\) | \(h'(1)=\frac{1}{2}>0\) | Increasing |
Check: The domain begins at \(0\), so the chart correctly has no interval to the left. For every \(x>0\), both \(2\) and \(\sqrt{x}\) are positive, so \(\frac{1}{2\sqrt{x}}>0\), confirming that the test-point sign holds throughout \((0,\infty)\).
Why One Test Point Can Represent an Interval
A sign chart would not be useful if a derivative could be positive at the chosen test point and negative a moment later without anything marking that change. On an interval where \(f'\) is continuous, it cannot change from positive to negative without taking the value zero somewhere in between. This follows directly from the Intermediate Value Theorem.
Indeed, suppose \(f'\) is continuous on an interval and takes a positive value at \(a\) and a negative value at \(b\), with \(a<b\). Since \(0\) lies between \(f'(a)\) and \(f'(b)\), the Intermediate Value Theorem gives a point \(c\) between \(a\) and \(b\) such that \(f'(c)=0\). Consequently, an interval with no derivative zeros cannot contain both positive and negative derivative values when \(f'\) is continuous there. This is why critical numbers must be included among the partition points.
Once the sign is known throughout an interval, the earlier result on derivative signs and monotonicity gives the behavior row: a positive derivative means increasing, and a negative derivative means decreasing. The chart is therefore built in two stages: establish the derivative sign on each interval, then translate that sign into monotonicity.
Common Pitfalls
Testing at a critical number. A critical number is a boundary between intervals, not a representative point inside one. For the cubic example, testing \(x=1\) to decide the sign on \((-1,1)\) would give \(f'(1)=0\). That tells us the derivative vanishes at the boundary, not whether it is positive or negative throughout the interval. The correction is to choose a point strictly inside, such as \(x=0\), and compute \(f'(0)=-3\).
Calling an excluded point critical. Critical numbers must belong to the domain of the function. If a formula is undefined at a point, first check the function’s domain; a point excluded from that domain is not a critical number. It may still affect which domain intervals you can analyze, but deciding how to build charts over disconnected domains requires extra care.
Skipping a critical number. If even one zero or in-domain point where the derivative does not exist is omitted, an interval in the chart may contain a place where the derivative’s sign changes. Solve \(f'(x)=0\) completely and check separately for points in the domain where the derivative does not exist before selecting test points.
Writing behavior at a point instead of on an interval. The sign \(f'(c)=0\) at a critical number does not say whether the function increases or decreases on either side. Record monotonicity on the intervals, and keep the critical numbers in their own boundary row or column.
Why the Chart Is Useful
A sign chart condenses a calculation into a form that is easy to inspect and use. It makes omissions visible: the ordered critical numbers show how the domain was partitioned, while the test values document why each interval received its sign. The increasing and decreasing rows then provide the information needed for later questions about a function’s behavior. In particular, the chart records sign patterns without requiring us to interpret a change in sign as a particular kind of extremum.
Before accepting a chart, check that every critical number belongs to the domain, the numbers are ordered correctly, each interval has a valid test point, and every derivative sign matches the resulting monotonicity statement. These checks make the chart a dependable summary rather than a collection of guesses.
Check Your Understanding
For each question, explain your reasoning and show any derivative evaluations needed.
- For \(p(x)=x^3-12x\), find the critical numbers and state the intervals that should appear in a sign chart.
- For \(p(x)=x^3-12x\), use one test point in each interval to determine the signs of \(p'\) and the increasing or decreasing behavior.
- Why must a test point lie strictly inside the interval it represents?
- For \(q(x)=\sqrt{x}\), why is \(0\) a critical number even though it is an endpoint of the domain?
- State the condition that makes it valid to use one derivative test value to represent the sign throughout an interval.
