The Central Idea
If
\[ y=f(x), \]then its derivative
\[ f'(x) \]can be viewed as a slope function.
For each input \(x\) where the derivative exists, \(f'(x)\) tells us the slope of the tangent line to the graph of \(f\).
\[ \boxed{ x \longmapsto f'(x) \longmapsto \text{local slope of }f } \]Special inputs occur where that slope is zero or where the derivative fails to exist. These are the main candidates for critical behavior.
The Derivative as a New Function
Consider
\[ f(x)=x^2. \]Its derivative is
\[ f'(x)=2x. \]The original function gives curve heights:
\[ f(-2)=4, \qquad f(0)=0, \qquad f(2)=4. \]The derivative gives tangent slopes:
\[ f'(-2)=-4, \qquad f'(0)=0, \qquad f'(2)=4. \]The two functions contain different information.
A Slope Table
For \(f(x)=x^2\), we can compare function values and derivative values.
| \(x\) | \(f(x)=x^2\) | \(f'(x)=2x\) | Local slope |
|---|---|---|---|
| \(-2\) | \(4\) | \(-4\) | Negative |
| \(-1\) | \(1\) | \(-2\) | Negative |
| \(0\) | \(0\) | \(0\) | Horizontal |
| \(1\) | \(1\) | \(2\) | Positive |
| \(2\) | \(4\) | \(4\) | Positive |
The derivative table acts like a numerical map of the curve's local direction.
Positive Slope-Function Values
If
\[ f'(x)>0, \]then the tangent line at that input has positive slope.
Locally, the graph rises as \(x\) increases.
For example, if
\[ f'(3)=7, \]then the tangent line to \(f\) at \(x=3\) has slope
\[ \boxed{7}. \]Negative Slope-Function Values
If
\[ f'(x)<0, \]the tangent line has negative slope.
Locally, the graph falls as \(x\) increases.
Thus,
\[ f'(-2)=-5 \]tells us that the graph is locally descending at \(x=-2\).
Zero Slope-Function Values
If
\[ f'(c)=0, \]then the tangent line at \(x=c\) is horizontal.
Such an input is often important because a curve may change from increasing to decreasing or from decreasing to increasing there.
But a zero derivative does not guarantee such a change.
What Is a Critical Number?
A critical number of a function \(f\) is an input \(c\) in the domain of \(f\) for which
\[ \boxed{ f'(c)=0 } \]or for which
\[ \boxed{ f'(c)\text{ does not exist}. } \]Critical numbers identify inputs where ordinary tangent behavior may be especially important.
What Is a Critical Point?
A critical number is an input value.
A critical point is the corresponding point on the graph.
If \(c\) is a critical number, then the associated critical point is
\[ \boxed{ (c,f(c)) }. \]Keeping the input and the graph point distinct prevents a common vocabulary error.
Example: Critical Number versus Critical Point
Let
\[ f(x)=x^2-4. \]Differentiate:
\[ f'(x)=2x. \]Set the derivative equal to zero:
\[ 2x=0. \]Thus,
\[ \boxed{x=0} \]is the critical number.
Since
\[ f(0)=-4, \]the critical point is
\[ \boxed{(0,-4)}. \]Why Critical Numbers Matter
Critical numbers are places where the ordinary slope information changes character.
They may correspond to:
- horizontal tangents;
- local maxima;
- local minima;
- flat points that are not extrema;
- corners;
- cusps;
- vertical tangent behavior;
- other points where the derivative is undefined.
A critical number is therefore a candidate for important behavior, not a complete conclusion by itself.
Critical Numbers of Polynomials
Polynomial functions are differentiable for every real input.
Therefore, for a polynomial, the derivative never fails to exist.
Its critical numbers come from solving
\[ \boxed{ f'(x)=0 }. \]This makes polynomial critical-number problems especially direct.
Example: A Quadratic Critical Number
Let
\[ f(x)=x^2-6x+5. \]Differentiate:
\[ f'(x)=2x-6. \]Solve
\[ 2x-6=0. \]Then
\[ 2x=6 \]and
\[ \boxed{x=3}. \]Thus, \(x=3\) is the critical number.
Finding the Critical Point
Continue with
\[ f(x)=x^2-6x+5. \]The critical number is
\[ x=3. \]Evaluate the original function:
\[ f(3) = 9-18+5 = -4. \]Therefore, the critical point is
\[ \boxed{(3,-4)}. \]The Derivative Graph and the Original Graph
The graph of \(f'(x)\) can be interpreted as a graph of the slopes of \(f(x)\).
If the derivative graph lies above the \(x\)-axis, then
\[ f'(x)>0. \]If it lies below the \(x\)-axis, then
\[ f'(x)<0. \]Where it crosses or touches the \(x\)-axis,
\[ f'(x)=0. \]Those zeros are critical-number candidates for the original function.
Zeros of the Derivative
Solving
\[ f'(x)=0 \]means finding the \(x\)-intercepts of the derivative function.
Thus, critical-number analysis creates a useful connection:
\[ \boxed{ \text{zero of }f' \longleftrightarrow \text{horizontal tangent of }f }. \]Example: A Cubic with Two Critical Numbers
Let
\[ f(x)=x^3-3x^2. \]Differentiate:
\[ f'(x)=3x^2-6x. \]Factor:
\[ f'(x)=3x(x-2). \]Set the derivative equal to zero:
\[ 3x(x-2)=0. \]By the zero-product principle,
\[ x=0 \]or
\[ x=2. \]Therefore, the critical numbers are
\[ \boxed{0,\ 2}. \]Critical Points of the Cubic
For
\[ f(x)=x^3-3x^2, \]the critical numbers are \(0\) and \(2\).
Evaluate:
\[ f(0)=0 \]and
\[ f(2)=8-12=-4. \]Therefore, the critical points are
\[ \boxed{(0,0)} \]and
\[ \boxed{(2,-4)}. \]Critical Numbers Can Come from Factoring
Polynomial derivatives frequently factor into simpler expressions.
For example,
\[ f'(x)=4x^3-16x \]factors as
\[ 4x(x^2-4) \]and then as
\[ 4x(x-2)(x+2). \]Setting this equal to zero gives
\[ \boxed{x=-2,\ 0,\ 2}. \]Familiar algebraic factoring remains an important calculus tool.
Example: Three Critical Numbers
Let
\[ f(x)=x^4-8x^2. \]Differentiate:
\[ f'(x)=4x^3-16x. \]Factor:
\[ f'(x)=4x(x^2-4) \] \[ =4x(x-2)(x+2). \]Therefore,
\[ \boxed{x=-2,\ 0,\ 2} \]are the critical numbers.
Finding All Three Critical Points
For
\[ f(x)=x^4-8x^2, \]evaluate at the critical numbers.
At \(x=-2\),
\[ f(-2)=16-32=-16. \]At \(x=0\),
\[ f(0)=0. \]At \(x=2\),
\[ f(2)=16-32=-16. \]Thus, the critical points are
\[ \boxed{(-2,-16),\ (0,0),\ (2,-16)}. \]A Critical Number Need Not Be an Extremum
Consider
\[ f(x)=x^3. \]Then
\[ f'(x)=3x^2. \]The derivative is zero at
\[ x=0. \]Thus, \(0\) is a critical number.
However, the graph continues increasing through the origin. The point
\[ (0,0) \]is not a local maximum or minimum.
Stationary Points
A point where
\[ f'(c)=0 \]is often called a stationary point.
Every stationary point is associated with a critical number under the definition used here.
But not every critical point must be stationary, because a derivative can also fail to exist.
Critical Behavior Where the Derivative Is Undefined
Critical numbers may also occur where the function exists but its derivative does not.
Typical examples include:
- corners;
- cusps;
- certain vertical tangents.
These cases are not polynomial examples, but they are important to the general definition.
Example: A Corner
Consider
\[ f(x)=|x|. \]The function is defined at
\[ x=0. \]But the slope approaching from the left is
\[ -1, \]while the slope approaching from the right is
\[ 1. \]There is no single derivative at the origin.
Therefore,
\[ \boxed{x=0} \]is a critical number because \(f\) is defined there but \(f'(0)\) does not exist.
The Function Must Be Defined at a Critical Number
The requirement that \(c\) belong to the domain of \(f\) matters.
Suppose a function is undefined at \(x=2\).
Then \(x=2\) is not a critical number of that function under the standard definition, even if the derivative formula is also undefined there.
A critical number must correspond to an actual input of the original function.
Critical Numbers and Domain Restrictions
For nonpolynomial functions, a reliable procedure is:
- Determine where \(f\) is defined.
- Find \(f'(x)\) where possible.
- Solve \(f'(x)=0\).
- Identify inputs in the domain of \(f\) where \(f'(x)\) is undefined.
- Combine the valid candidates.
For polynomials, the first and fourth steps are automatic because both the function and derivative are defined for all real \(x\).
Critical Numbers Are Candidate Locations
A critical number identifies a place that deserves attention.
It does not by itself tell us exactly what happens there.
To classify the behavior, we may need to examine:
- the sign of \(f'(x)\) on nearby intervals;
- nearby function values;
- the shape of the graph;
- later, the second derivative.
These methods are developed more fully in the next lesson.
A First Look at Sign Changes
Suppose \(c\) is a critical number.
If
\[ f'(x)>0 \]just before \(c\), but
\[ f'(x)<0 \]just after \(c\), then the graph changes from rising to falling.
This pattern suggests a local maximum.
We will formalize this reasoning in the next lesson.
Another Sign-Change Pattern
If
\[ f'(x)<0 \]before a critical number and
\[ f'(x)>0 \]after it, the graph changes from falling to rising.
This pattern suggests a local minimum.
No Sign Change
A zero derivative may occur without changing the sign of the slope function.
For
\[ f(x)=x^3, \]we have
\[ f'(x)=3x^2. \]The derivative is positive on both sides of \(0\), even though
\[ f'(0)=0. \]The function keeps increasing through the critical point.
Multiplicity in the Derivative Can Matter
Consider a derivative written in factored form:
\[ f'(x)=(x-2)^2(x+1). \]The critical numbers are
\[ x=2 \]and
\[ x=-1. \]The squared factor
\[ (x-2)^2 \]does not change sign as \(x\) passes through \(2\).
This can influence whether the original function changes from increasing to decreasing there.
Factored Derivatives Are Often Easier to Analyze
Suppose
\[ f'(x)=x^2-5x+6. \]Factoring gives
\[ f'(x)=(x-2)(x-3). \]The zeros are immediately visible:
\[ \boxed{x=2,\ 3}. \]Factored form also makes later sign analysis easier.
Example: Derivative Already Given
Suppose
\[ f'(x)=2(x+3)(x-1). \]To find critical numbers, set
\[ 2(x+3)(x-1)=0. \]Therefore,
\[ x=-3 \]or
\[ x=1. \]If \(f\) is defined at both inputs, then
\[ \boxed{-3,\ 1} \]are critical numbers.
Critical Values
If \(c\) is a critical number, the corresponding output
\[ f(c) \]is sometimes called a critical value.
Thus, for a critical point
\[ (c,f(c)), \]the three related objects are:
- critical number: \(c\);
- critical value: \(f(c)\);
- critical point: \((c,f(c))\).
Example: Number, Value, and Point
Let
\[ f(x)=x^2-4x+7. \]Differentiate:
\[ f'(x)=2x-4. \]Solve:
\[ 2x-4=0 \] \[ x=2. \]The critical number is
\[ \boxed{2}. \]The critical value is
\[ f(2)=4-8+7=3. \]The critical point is
\[ \boxed{(2,3)}. \]Critical Numbers from a Slope Table
Suppose the following derivative values are known:
| \(x\) | \(f'(x)\) |
|---|---|
| \(-2\) | \(5\) |
| \(-1\) | \(2\) |
| \(0\) | \(0\) |
| \(1\) | \(-2\) |
| \(2\) | \(-5\) |
The table directly identifies
\[ \boxed{x=0} \]as a critical-number candidate because the derivative equals zero there.
Critical Numbers from a Graph of \(f'\)
If the graph of the derivative is available, look for inputs where:
- the graph of \(f'\) meets the \(x\)-axis;
- the graph of \(f'\) has a hole or break while \(f\) remains defined;
- the derivative is otherwise undefined at an input belonging to \(f\)'s domain.
These inputs are candidates for critical numbers of the original function.
The Original Graph and Horizontal Tangents
If we are given the graph of \(f\) rather than \(f'\), a visible horizontal tangent suggests
\[ f'(c)=0. \]Thus, points where the graph appears locally flat are natural places to inspect for critical numbers.
Graphical estimates should be treated as approximate unless exact information is provided.
Polynomial Critical Numbers May Be Irrational
Solving \(f'(x)=0\) does not always produce integers or rational numbers.
Suppose
\[ f'(x)=3x^2-6. \]Set equal to zero:
\[ 3x^2-6=0. \]Then
\[ x^2=2, \]so
\[ \boxed{x=\pm\sqrt{2}}. \]Both are valid critical numbers if \(f\) is a polynomial.
Polynomial Critical Numbers May Require the Quadratic Formula
Suppose
\[ f'(x)=2x^2+3x-1. \]This quadratic does not factor conveniently over the integers.
Use the quadratic formula:
\[ x = \frac{-3\pm\sqrt{3^2-4(2)(-1)}}{2(2)}. \]Therefore,
\[ \boxed{ x=\frac{-3\pm\sqrt{17}}{4} }. \]Calculus often depends on algebraic solution methods already developed earlier in Malone Math.
Critical Numbers and Restricted Intervals
Sometimes a problem asks about a function only on a specified interval.
Suppose the critical-number equation gives
\[ x=-3,\ 1,\ 5, \]but the problem concerns
\[ 0\le x\le4. \]Only
\[ \boxed{x=1} \]lies inside the stated interval.
Interval restrictions must be applied after solving.
Endpoints Deserve Separate Attention
When analyzing a function on a closed interval, endpoints can matter even when they are not treated the same way as interior stationary points.
For example, on
\[ [a,b], \]the values
\[ f(a) \]and
\[ f(b) \]may need to be compared with values at interior critical numbers.
This becomes especially important in optimization.
Not Every Zero of a Formula Is a Critical Number
The equation that matters is
\[ f'(x)=0, \]not merely
\[ f(x)=0. \]Zeros of \(f\) are \(x\)-intercepts of the original graph.
Zeros of \(f'\) identify horizontal tangent candidates.
These are different kinds of information.
Example: Function Zeros versus Derivative Zeros
Let
\[ f(x)=x^2-4. \]The function zeros satisfy
\[ x^2-4=0, \]so
\[ x=\pm2. \]But
\[ f'(x)=2x, \]whose zero is
\[ x=0. \]Therefore, the \(x\)-intercepts and critical number are not the same.
Common Error: Solving \(f(x)=0\) Instead of \(f'(x)=0\)
To find stationary critical numbers, first differentiate.
The correct sequence is
\[ f(x) \longrightarrow f'(x) \longrightarrow f'(x)=0. \]Solving
\[ f(x)=0 \]finds zeros of the original function, not critical numbers in general.
Common Error: Reporting Only the Critical Point When Asked for the Number
If the critical number is
\[ x=2 \]and
\[ f(2)=5, \]then:
- critical number: \(2\);
- critical point: \((2,5)\).
Match the requested object to the requested terminology.
Common Error: Assuming Every Critical Point Is a Maximum or Minimum
A critical point is a candidate for special behavior.
For
\[ f(x)=x^3, \]the origin is critical because
\[ f'(0)=0. \]But the graph is increasing on both sides of the origin.
Classification requires additional analysis.
Common Error: Ignoring Undefined Derivatives
The equation
\[ f'(x)=0 \]finds stationary critical numbers, but the full definition also includes domain points where
\[ f'(x) \]does not exist.
Corners and cusps are important examples.
Common Error: Including Inputs Outside the Domain
If \(f\) is not defined at \(x=c\), then \(c\) is not a critical number of \(f\) under the definition used here.
Always check the domain when working with rational, radical, piecewise, or other nonpolynomial functions.
Common Error: Forgetting to Solve the Derivative Equation Completely
Suppose
\[ f'(x)=x(x-3)(x+4). \]There are three zeros:
\[ x=0, \qquad x=3, \qquad x=-4. \]Stopping after finding only one factor's zero would miss critical numbers.
A Reliable Procedure for Polynomial Critical Numbers
- Differentiate the polynomial.
- Set \(f'(x)=0\).
- Factor the derivative when possible.
- Use the zero-product principle or another algebraic solution method.
- Keep every real solution allowed by the stated interval.
- Report those inputs as critical numbers.
- If critical points are requested, evaluate \(f(c)\) for each critical number.
A Reliable Procedure for General Critical Numbers
- Determine the domain of \(f\).
- Find \(f'(x)\) wherever possible.
- Solve \(f'(x)=0\).
- Find domain inputs where \(f'(x)\) is undefined.
- Discard any candidate not in the domain of \(f\).
- Apply any interval restrictions.
- Evaluate the original function if critical points are needed.
Example: Full Polynomial Workflow
Find the critical numbers and critical points of
\[ f(x)=x^3-6x^2+9x+2. \]Differentiate:
\[ f'(x)=3x^2-12x+9. \]Factor:
\[ f'(x)=3(x^2-4x+3) \] \[ =3(x-1)(x-3). \]Set equal to zero:
\[ 3(x-1)(x-3)=0. \]Therefore, the critical numbers are
\[ \boxed{x=1,\ 3}. \]Completing the Full Polynomial Workflow
Evaluate the original function.
At \(x=1\),
\[ f(1) = 1-6+9+2 = 6. \]At \(x=3\),
\[ f(3) = 27-54+27+2 = 2. \]Therefore, the critical points are
\[ \boxed{(1,6),\ (3,2)}. \]Example: A Repeated Critical Number
Let
\[ f(x)=x^4. \]Differentiate:
\[ f'(x)=4x^3. \]Solve:
\[ 4x^3=0. \]Thus,
\[ \boxed{x=0} \]is the only critical number.
A repeated factor in the derivative does not create multiple distinct critical numbers at the same input.
Example: No Critical Numbers
Consider
\[ f(x)=2x+7. \]Then
\[ f'(x)=2. \]Since
\[ 2\ne0 \]and the derivative exists everywhere, the function has
\[ \boxed{\text{no critical numbers}}. \]A nonhorizontal linear function has no stationary or undefined-derivative points.
Example: Every Input of a Constant Function
Consider
\[ f(x)=5. \]Then
\[ f'(x)=0 \]for every real \(x\).
Under the definition used here, every real input is a critical number.
This reminds us that critical does not necessarily mean isolated or dramatic. It means that the derivative satisfies the critical-number condition.
Practice 1: Slope Function
Let
\[ f(x)=3x^2-4x+1. \]Find the slope function \(f'(x)\).
Practice 2: Evaluate the Slope Function
For
\[ f(x)=3x^2-4x+1, \]find the tangent slope at \(x=2\).
Practice 3: One Critical Number
Find the critical number of
\[ f(x)=x^2+8x+3. \]Practice 4: Critical Point
For
\[ f(x)=x^2+8x+3, \]find the corresponding critical point.
Practice 5: Two Critical Numbers
Find the critical numbers of
\[ f(x)=x^3-12x. \]Practice 6: Three Critical Numbers
Find the critical numbers of
\[ f(x)=x^4-10x^2. \]Practice 7: No Critical Numbers
Determine whether
\[ f(x)=-5x+9 \]has any critical numbers.
Practice 8: Critical Number from an Undefined Derivative
The function
\[ f(x)=|x-2| \]has a corner at \(x=2\).
Explain why \(x=2\) is a critical number.
Practice 9: Critical Number versus Function Zero
Let
\[ f(x)=x^2-9. \]Find the zeros of \(f\), then find the critical number. Explain the difference.
Practice 10: Identify the Error
A student is given
\[ f(x)=x^3-3x \]and solves
\[ x^3-3x=0 \]to find the critical numbers.
Explain the error and find the correct critical numbers.
Practice Answers
Practice 1
\[ f'(x)=6x-4. \]Therefore,
\[ \boxed{f'(x)=6x-4}. \]Practice 2
\[ f'(2)=6(2)-4=8. \]The tangent slope is
\[ \boxed{8}. \]Practice 3
\[ f'(x)=2x+8. \]Set equal to zero:
\[ 2x+8=0. \] \[ x=-4. \]Therefore, the critical number is
\[ \boxed{-4}. \]Practice 4
The critical number is \(x=-4\). Evaluate:
\[ f(-4)=16-32+3=-13. \]Therefore, the critical point is
\[ \boxed{(-4,-13)}. \]Practice 5
\[ f'(x)=3x^2-12. \]Set equal to zero:
\[ 3x^2-12=0. \] \[ x^2=4. \]Therefore,
\[ \boxed{x=-2,\ 2}. \]Practice 6
\[ f'(x)=4x^3-20x. \]Factor:
\[ 4x(x^2-5)=0. \]Therefore,
\[ x=0 \]or
\[ x^2=5. \]Thus, the critical numbers are
\[ \boxed{x=-\sqrt5,\ 0,\ \sqrt5}. \]Practice 7
\[ f'(x)=-5. \]The derivative is never zero and never undefined, so the function has
\[ \boxed{\text{no critical numbers}}. \]Practice 8
The function \(f(x)=|x-2|\) is defined at \(x=2\), but the corner prevents a single tangent slope from existing there.
Therefore,
\[ \boxed{x=2} \]is a critical number because \(f'(2)\) does not exist.
Practice 9
The zeros of
\[ f(x)=x^2-9 \]satisfy
\[ x^2-9=0, \]so
\[ \boxed{x=-3,\ 3}. \]For the critical number,
\[ f'(x)=2x. \]Then
\[ 2x=0 \]gives
\[ \boxed{x=0}. \]The zeros locate \(x\)-intercepts of \(f\); the critical number locates a horizontal tangent.
Practice 10
The student solved \(f(x)=0\), which finds zeros of the original function.
Instead,
\[ f'(x)=3x^2-3. \]Set the derivative equal to zero:
\[ 3x^2-3=0. \] \[ x^2=1. \]Therefore, the critical numbers are
\[ \boxed{x=-1,\ 1}. \]What to Remember
The derivative acts as a slope function:
\[ \boxed{ f'(x) = \text{local slope of }f\text{ at }x }. \]A critical number \(c\) is an input in the domain of \(f\) where
\[ \boxed{ f'(c)=0 \quad\text{or}\quad f'(c)\text{ does not exist} }. \]- positive derivative values correspond to positive tangent slopes;
- negative derivative values correspond to negative tangent slopes;
- zero derivative values correspond to horizontal tangents;
- for polynomials, critical numbers are found by solving \(f'(x)=0\);
- a critical number is an input \(c\);
- a critical value is the corresponding output \(f(c)\);
- a critical point is \((c,f(c))\);
- a critical point is a candidate for important behavior, not automatically a maximum or minimum;
- domain restrictions and interval restrictions must be respected.
Where We Go Next
Critical numbers divide a curve into regions where the slope function may have different signs. Those sign patterns let us determine where the original function rises or falls and help classify local maxima and minima. A second derivative then gives another layer of information about how the slopes themselves are changing.
The next lesson, Increasing, Decreasing, Extrema, and Concavity, develops these ideas systematically and uses first- and second-derivative information to analyze the shape of a curve.