Topic 1.4 · CED: Polynomial Functions and Rates of Change

Extrema and Rates of Change

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01
A polynomial is a finite sum of terms $a x^n$ where every exponent $n$ is a whole number ( $0, 1, 2, \dots$ ) and every coefficient is a real number.
02
The degree is the largest exponent that has a non-zero coefficient.

The leading coefficient is the number multiplying that highest-power term.
03
Degree of a product $=$ sum of the degrees.

Leading coefficient of a product $=$ product of the leading coefficients.

A polynomial is a finite sum of terms $a x^n$ where every exponent $n$ is a whole number ( $0, 1, 2, \dots$ ) and every coefficient is a real number.
The degree is the largest exponent that has a non-zero coefficient.

The leading coefficient is the number multiplying that highest-power term.
Degree of a product $=$ sum of the degrees.

Leading coefficient of a product $=$ product of the leading coefficients.
Local max: increasing → decreasing. Local min: decreasing → increasing.
If a polynomial has two zeros, there must be at least one extremum between them.
Maximum turning points = degree − 1
An inflection point is where the graph switches between concave up and concave down.

At an inflection point the direction does not have to change — the function can keep rising the whole way through.
A polynomial of degree $n$ has at most $n - 1$ turning points and at most $n - 2$ inflection points.

Practice 24 problems Next: Chapter 1.5A Polynomial Functions and Complex Zeros

What Counts as a Polynomial

Every rule in this topic is a rule about polynomials, so it is worth being exact about what one is.

KEY RULE

A polynomial is a finite sum of terms $a x^n$ where every exponent $n$ is a whole number ( $0, 1, 2, \dots$ ) and every coefficient is a real number.

CONCEPT

What Is NOT a Polynomial

$\sqrt{x} = x^{1/2}$ — the exponent is not a whole number.

$\dfrac{1}{x} = x^{-1}$ — the exponent is negative.

$2^x$ — the variable is in the exponent, not the base.

$|x|$ — absolute value is not a power of $x$ .

Degree and Leading Coefficient

KEY RULE

The DEGREE is the largest exponent that has a non-zero coefficient.

The LEADING COEFFICIENT is the number multiplying that highest-power term.

Write the polynomial in descending order first and both are easy to read.

Worked example

Give the degree and leading coefficient of $p(x) = 7 - 4x^3 + x^2$ .

01
Put the terms in descending order of exponent:
$p(x) = -4x^3 + x^2 + 7$
02
The largest exponent is $3$ , so the degree is $3$ .
03
The number in front of $x^3$ is $-4$ , so the leading coefficient is $-4$ .

COMMON MISTAKE

Reading the degree off the term written first. In $7 - 4x^3 + x^2$ the first term is the constant — the degree is still $3$ .

Forgetting the sign of the leading coefficient. It is $-4$ , not $4$ , and that sign decides the end behavior.

Products of Polynomials

KEY RULE

Degree of a product $=$ sum of the degrees.

Leading coefficient of a product $=$ product of the leading coefficients.

So $(-2x^3 + 1)(5x^2 - x)$ has degree $3 + 2 = 5$ and leading coefficient $(-2)(5) = -10$ — no expansion needed. Topic 1.6 uses this constantly.

Quick check

What are the degree and leading coefficient of $7 - 4x^{3} + x^{2}$ ?

What Is a Local Maximum?

A local maximum is a point where the function is increasing right before it, and decreasing right after it — like the top of a hill.

CONCEPT

Vocabulary Note

"Local max" and "relative max" mean the EXACT SAME THING. You'll see both terms used — textbooks and teachers switch between them freely.

The plural of "maximum" is "maxima" (so "local maxima" = more than one local max point).

Same pattern for minimum → minima.

The function increases, peaks, then decreases — that peak is the local max.

What Is a Local Minimum?

A local minimum is the mirror idea — a point where the function is decreasing right before it, and increasing right after it — like the bottom of a valley.

The function decreases, bottoms out, then increases — that valley is the local min.

KEY RULE

Local max: increasing → decreasing. Local min: decreasing → increasing.

Global (Absolute) Extrema

A function can have several local maxima and local minima. Among all of them, the single tallest local max is called the global (absolute) maximum, and the single lowest local min is called the global (absolute) minimum.

CONCEPT

Definition

The global (absolute) maximum is the highest output the function reaches anywhere in its domain — the tallest of its local peaks.

The global (absolute) minimum is the lowest output it reaches anywhere — the deepest of its local valleys.

A function can have one, both, or neither (if it rises or falls without bound).

Four local extrema total — but only the tallest peak and lowest valley are 'absolute'.

COMMON MISTAKE

Every absolute max/min IS a local max/min — but not every local max/min is absolute.

A function is not required to have an absolute max or min — if the graph goes to infinity in some direction, that extreme may not exist (see Section 5).

Endpoints Can Be Extrema Too

If a function's domain has a restricted (included) endpoint — shown on a graph as a closed/filled dot — that endpoint can itself be a local extremum, even though the curve doesn't turn around there in the usual increasing/decreasing sense.

CONCEPT

Reading Graph Symbols

Closed (filled) dot → that point IS included in the domain — it can be an extremum.

Open circle → that point is NOT included — a value gets arbitrarily close but never reaches it.

Arrow → the graph keeps going forever in that direction (unbounded).

Step-by-Step: Reading Extrema from a Graph

Worked example

For $f(x) = x^{3} - 3x$ on the domain $[-3, \infty)$ , find the absolute min, absolute max, and all local extrema.

01
Scan for local extrema first: there's a local max at $x=-1$ (value 2), and a local min at $x=1$ (value −2).
02
Check the left end: $x=-3$ is a CLOSED dot (included), with value $f(-3) = (-3)^{3}-3(-3) = -27+9 = -18$ . This is much lower than the local min of −2.
03
Since $-18$ is the lowest value anywhere on the graph, the ABSOLUTE MIN is $-18$ at $x = -3$ — it happens at the closed endpoint, not at the interior local min of $-2$ .
04
Check the right end: the arrow shows the graph increasing forever as $x \to \infty$ . Since it never stops climbing, there is NO highest value.
05
ABSOLUTE MAX = NONE (does not exist). Final answers — Absolute min: −18 at $x=-3$ . Absolute max: NONE. Local max: 2 at $x=-1$ . Local min: −2 at $x=1$ .

CONCEPT

Reading Strategy Checklist

Find all local maxima and minima by looking for peaks and valleys.
Check BOTH endpoints — if a dot is closed, evaluate the function there and compare to the local extrema.
If an arrow shows the graph continuing to $+\infty$ or $-\infty$ forever, that absolute extremum does NOT exist — write NONE.

The Two-Zeros Rule

KEY RULE

If a polynomial has two zeros, there must be at least ONE extremum between them.

Between any two zeros of a polynomial, the graph must turn around at least once.

Why? To get from $f(x)=0$ back to $f(x)=0$ without immediately crossing the x-axis again, the function has to turn around somewhere in between — and that turning point is a local extremum.

Even Degree Polynomials Always Have an Absolute Extremum

Remember end behavior: even-degree polynomials have both ends pointing the same direction.

CONCEPT

The Rule

On the full domain (all real numbers):

Even degree + positive leading coefficient → both ends go UP → there IS an absolute minimum (but no absolute max).

Even degree + negative leading coefficient → both ends go DOWN → there IS an absolute maximum (but no absolute min).

Odd-degree polynomials never have an absolute max or min — the ends go in opposite directions, off to $+\infty$ and $-\infty$ .

Restrict the domain and this changes: on $[-3, \infty)$ the odd-degree $f(x) = x^3 - 3x$ does have an absolute min, at the closed endpoint. Section 5 works that case.

Turning Points and Rate of Change

A cubic (degree 3) polynomial's rate of change can speed up and slow down, even reversing direction — this is exactly what creates local maxima and minima.

KEY RULE

Maximum turning points = degree − 1

A cubic (degree 3) can have up to 2 turning points; a quartic (degree 4) can have up to 3.

Quick check

A polynomial has degree $5$ . What is the greatest number of turning points its graph can have?

Step-by-Step: Detecting a Turning Point via Sign Change

Worked example

For $f(x) = x^3 - 3x$ , is $f$ increasing or decreasing near $x = 0$ and near $x = 1.5$ ?

01
Near $x = 0$ , check $[-0.5, 0.5]$ :
$f(-0.5) = (-0.5)^3 - 3(-0.5) = -0.125 + 1.5 = 1.375$ $f(0.5) = (0.5)^3 - 3(0.5) = 0.125 - 1.5 = -1.375$
02
Average rate of change:
$\frac{-1.375 - 1.375}{0.5 - (-0.5)} = \frac{-2.75}{1} = -2.75$
03
Negative → over this interval $f$ ends up lower than it started. (Careful: an average rate of change does not say what happens at every point in between — see 1.2.)
04
Near $x = 1.5$ , check $[1, 2]$ :
$f(1) = 1 - 3 = -2 \qquad f(2) = 8 - 6 = 2$
05
Average rate of change:
$\frac{2 - (-2)}{2 - 1} = \frac{4}{1} = 4$
06
Positive → over $[1, 2]$ , $f$ ends up higher than it started. The sign flipped between the two intervals, so a turning point lies somewhere in $[-0.5, 2]$ . (It is at $x = 1$ — the left endpoint of the second interval, not strictly between the two.)

COMMON MISTAKE

Don't confuse 'local' with 'absolute' — always check whether the question is asking for one specific peak/valley or the single tallest/lowest.

A closed endpoint can be an absolute extremum even if the function never 'turns around' there.

Odd-degree polynomials never have an absolute max or min ON THEIR FULL DOMAIN. If the domain is restricted, check the closed endpoints — one of them can be the absolute extremum.

Inflection Points and the Rate of Change

A turning point is where the function changes DIRECTION. An inflection point is where it changes the way it BENDS. They are different questions and the exam asks both.

KEY RULE

An INFLECTION POINT is where the graph switches between concave up and concave down.

At an inflection point the direction does NOT have to change — the function can keep rising the whole way through.

CONCEPT

What Happens to the Rate of Change There

Concave up means the rates of change are increasing. Concave down means they are decreasing.

So at an inflection point the rates stop increasing and start decreasing (or the reverse). The rate of change reaches a local maximum or a local minimum exactly there.

In plain words: an inflection point is where the function is growing at its fastest — or its slowest — even though it is still growing.

How Many Can There Be

KEY RULE

A polynomial of degree $n$ has at most $n - 1$ turning points and at most $n - 2$ inflection points.

A cubic ( $n = 3$ ) has at most $2$ turning points and at most $1$ inflection point. A quadratic ( $n = 2$ ) has at most $1$ turning point and NO inflection points at all — a parabola bends the same way forever.

Worked example

For $f(x) = x^3 - 3x$ , the graph is concave down on $(-\infty, 0)$ and concave up on $(0, \infty)$ . Where is the inflection point, and what is happening to the rate of change there?

01
Concavity switches at $x = 0$ , so the inflection point is at $x = 0$ .
02
Evaluate to get the full point: $f(0) = 0 - 0 = 0$ , so it is $(0, 0)$ .
03
Left of $0$ the graph is concave down, so the rates of change are decreasing there. Right of $0$ it is concave up, so they are increasing.
04
The rates fall, then rise — so the rate of change hits its MINIMUM at $x = 0$ . The function is descending at its steepest exactly at the inflection point.

COMMON MISTAKE

Calling every inflection point a turning point. At $x = 0$ above, $f$ does not turn around — it is decreasing on both sides. Only the bending changes.

Looking for an inflection point on a parabola. There is never one.

Practice Problems

Worked example

Problem 1. For $f(x) = x^{3} - 12x$ , determine whether $f$ is increasing or decreasing on $[1, 2]$ .

01
Evaluate:
$f(1) = 1 - 12 = -11 \qquad f(2) = 8 - 24 = -16$
02
Average rate of change:
$\frac{-16 - (-11)}{2 - 1} = \frac{-5}{1} = -5$
03
Negative → over this interval $f$ ends up lower than it started. (An average rate of change is a net change, not a promise about every point in between.)

Worked example

Problem 2. A polynomial has degree 6 and a negative leading coefficient. Does it have an absolute max, absolute min, both, or neither?

01
Degree 6 is even, so both ends point the same direction.
02
Negative leading coefficient → both ends go DOWN → there is an ABSOLUTE MAX (no absolute min).

Common slips

Reading the degree off the term written first. In $7 - 4x^3 + x^2$ the first term is the constant — the degree is still $3$ .

Forgetting the sign of the leading coefficient. It is $-4$ , not $4$ , and that sign decides the end behavior.
Every absolute max/min is a local max/min — but not every local max/min is absolute.

A function is not required to have an absolute max or min — if the graph goes to infinity in some direction, that extreme may not exist (see Section 5).
Don't confuse 'local' with 'absolute' — always check whether the question is asking for one specific peak/valley or the single tallest/lowest.

A closed endpoint can be an absolute extremum even if the function never 'turns around' there.

Odd-degree polynomials never have an absolute max or min on their full domain. If the domain is restricted, check the closed endpoints — one of them can be the absolute extremum.
Calling every inflection point a turning point. At $x = 0$ above, $f$ does not turn around — it is decreasing on both sides. Only the bending changes.

Looking for an inflection point on a parabola. There is never one.

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3 cards · Rates of change

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Recap card

6 lines to re-read the night before.

01
Local max (= relative max): increasing then decreasing. Local min (= relative min): decreasing then increasing.
02
Absolute (global) extrema are the single tallest local max or single lowest local min — not every function has one.
03
Check endpoints: a closed dot can be the absolute extremum, even without a turn-around.
04
Two zeros guarantee at least one extremum between them.
05
On the full domain: even-degree polynomials always have an absolute max or min (never both); odd-degree polynomials have neither. A restricted domain changes this — check the closed endpoints.
06
Max turning points = degree − 1; a sign change in average rate of change signals a turning point nearby.

Practice 24 problems Next: Chapter 1.5A Polynomial Functions and Complex Zeros

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Extrema and Rates of Change

What Counts as a Polynomial

Degree and Leading Coefficient

Products of Polynomials

What Is a Local Maximum?

What Is a Local Minimum?

Global (Absolute) Extrema

Endpoints Can Be Extrema Too

Step-by-Step: Reading Extrema from a Graph

The Two-Zeros Rule

Even Degree Polynomials Always Have an Absolute Extremum

Turning Points and Rate of Change

Step-by-Step: Detecting a Turning Point via Sign Change

Inflection Points and the Rate of Change

How Many Can There Be

Practice Problems

Recap card

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