Application of Derivatives · Lesson 3 of 7

Understanding Maximum and Minimum Values

“Distinguish local and absolute extrema, recognise the role of the domain, and identify critical candidates.”

Learning Objectives

• Distinguish an extreme value from the input where it is attained. • Explain why the domain affects existence of maxima and minima. • Distinguish local extrema from extrema over an entire domain. • Identify stationary and nondifferentiable critical points. • Explain why a zero derivative is not enough to prove an extremum.

The Largest Value Depends on the Allowed Inputs

Suppose you want the greatest height of a ball, the largest possible profit, or the shortest distance to a curve. These questions are about extreme values. Before using any derivative, identify the quantity being compared and the set of inputs over which the comparison is allowed.

A function may have a largest value on one domain and no largest value on another. It may also reach its largest value at more than one input. An extreme value is an output; a point where it is attained is an input location. Keeping these two ideas separate makes the final answer precise.

Definition

Maximum and minimum on a domain

For inputs in a domain $I$ , $f (c)$ is a maximum value if $c \in I$ and $f (c) \geq f (x)$ for every $x \in I$ . It is a minimum value if $f (c) \leq f (x)$ for every $x \in I$ . Either is an extreme value. Equality is allowed, so several inputs can attain the same extreme value.

Use non-strict inequalities in these definitions. A largest output need not be uniquely attained: $sin x$ , for example, reaches the value 1 repeatedly. If every other value must be strictly smaller, we are discussing a strict maximum instead.

Example — A parabola on the whole real line

Problem
Find the maximum and minimum values of $f (x) = x^{2}$ for all real $x$ . Then restrict the domain to $[- 2, 1]$ .

1.Every square is nonnegative, and $f (0) = 0$ . Thus the minimum value is 0, attained at $x = 0$ .
2.On the whole real line there is no maximum: choosing $∣ x ∣$ large enough makes $x^{2}$ exceed any proposed bound.
3.On $[- 2, 1]$ , the least value is still 0. The greatest value is 4, attained at the included endpoint $x = - 2$ .
4.The formula did not change, but the permitted comparisons did. State the domain when giving an extreme value.

Changing the domain changes the maximum— On the displayed restricted domain [−2, 1], the largest value is the left endpoint value 4.

A Sharp Corner Can Still Be a Minimum

Differentiability is a condition for particular derivative tests, not a requirement for an extreme value to exist. A graph can have a lowest point at a sharp corner. Recognise such a point directly from function values before deciding how to test it with derivatives.

Example — The absolute-value graph

Problem
Find the extrema of $f (x) = ∣ x ∣$ on the real line, and then on $[- 2, 1]$ .

1.Since $∣ x ∣ \geq 0$ and $∣0∣ = 0$ , the minimum value is 0 at $x = 0$ . There is no maximum on the real line because $∣ x ∣$ is unbounded above.
2.On $[- 2, 1]$ , the maximum is 2 at $x = - 2$ , and the minimum remains 0 at $x = 0$ .
3.At zero the left slope is $- 1$ and the right slope is 1, so the derivative does not exist. The minimum is nevertheless well defined.

A minimum at a nondifferentiable point— The two straight pieces meet at a corner. Differentiability fails there, but all nearby values are larger.

Approaching a Bound Is Not Attaining It

A graph may get arbitrarily close to an output without ever reaching it. Such a bound is not an attained maximum or minimum. Open intervals make this distinction visible because the endpoint inputs are unavailable.

Example — No closest input to an excluded endpoint

Problem
Does $f (x) = x$ have a maximum or minimum on $(0, 1)$ ?

1.Take any allowed input $x_{0}$ . The input $x_{0} /2$ is also in $(0, 1)$ and gives a smaller output.
2.The input $(x_{0} + 1) /2$ is also in $(0, 1)$ and gives a larger output. No proposed value is least or greatest.
3.The function has neither maximum nor minimum on this open interval. Zero and one are bounds, but are not function values on this domain.
4.If the domain becomes $[0, 1]$ , the minimum is 0 at the left endpoint and the maximum is 1 at the right endpoint.

Excluded endpoints prevent attainment— The open circles are not in the graph of f on (0, 1). There is no first or last allowed input.

The role of a closed interval

A continuous function on a closed, bounded interval $[a, b]$ attains both a maximum and a minimum. A monotonic continuous function there attains them at endpoints. On an open or unbounded domain, attainment needs separate justification.

Local Peaks and Valleys

A landscape can contain a small hill below a distant mountain. Similarly, a local maximum is largest only among sufficiently nearby inputs. A local minimum is least only in a neighbourhood. It need not be the lowest value anywhere in the entire domain.

Definition

Local maximum

An interior domain point $c$ is a point of local maximum if some $h > 0$ can be chosen so that $f (c) \geq f (x)$ for every $x$ in $(c - h, c + h)$ . The output $f (c)$ is the local maximum value.

Definition

Local minimum

An interior domain point $c$ is a point of local minimum if some $h > 0$ satisfies $f (c) \leq f (x)$ for every $x$ in $(c - h, c + h)$ . The neighbourhood must lie inside the domain.

Local comparisons use a neighbourhood— The peak and valley are local extrema, but the cubic has neither an overall maximum nor an overall minimum on the real line.

Term	What is compared?	What is reported?
Local maximum	Values near an interior point	$c$ and $f (c)$
Local minimum	Values near an interior point	$c$ and $f (c)$
Absolute maximum	All values in the stated domain	Greatest attained output and all attaining inputs
Absolute minimum	All values in the stated domain	Least attained output and all attaining inputs

Local is not automatically absolute

A local maximum can be smaller than values far away. A function can have several local maxima of different heights. A derivative test near one input does not compare it with the whole domain.

Critical Points Are Candidates

At a smooth interior peak, a function changes from rising to falling. At a smooth valley, it changes from falling to rising. In both pictures the tangent is horizontal. At a sharp corner there may be no derivative, so searching only for horizontal tangents would miss possible extrema.

Definition

Critical point

A critical point is an input $c$ in the domain where either $f^{'} (c) = 0$ or the derivative does not exist. A stationary point is a critical point of the first kind, with zero derivative. A number outside the domain is not a critical point of the function.

Necessary condition at a smooth interior extremumLaTeX

f^{'} (c) = 0

f

has a local maximum or minimum at an interior point

c

and is differentiable there, its derivative is zero. Without differentiability, the derivative may not exist.

This is a necessary condition, not a complete classification. At a local maximum, right-hand difference quotients are nonpositive and left-hand difference quotients are nonnegative. If both approach the same derivative, that common value must be zero. At a local minimum the inequalities reverse, with the same conclusion.

The converse is false. For $f (x) = x^{3}$ , $f^{'} (0) = 0$ , but inputs just left of zero give negative outputs and inputs just right give positive outputs. The function keeps rising through zero. It is stationary there but has no maximum or minimum there.

A stationary point without a turning point— The curve keeps rising through (0, 0); a horizontal tangent alone does not guarantee an extremum.

The motivating applications now have a common structure: identify an output such as profit, height, or squared distance, establish its domain, and look for possible extreme points. The next lessons supply tests for candidates and procedures for comparing an entire domain.

Do not assume extra smoothness

Knowing that a function is continuous and has derivative zero at one point does not prove it is differentiable at every nearby point. Apply each derivative test only when its stated conditions are satisfied.

Check Your Understanding

Use these questions to check both your calculations and your interpretation of the ideas. Read the conditions carefully before choosing an answer.

Quick check

What is the minimum value of $x^{2}$ on the real line?

Quick check

Which statement about $f (x) = x$ on $(0, 1)$ is correct?

Quick check

Which point is critical for $∣ x ∣$ ?

Quick check

What does $f^{'} (c) = 0$ by itself tell us?

Quick check

On $[- 2, 1]$ , where does $x^{2}$ attain its absolute maximum?

Quick check

Which condition guarantees that both absolute extrema are attained?

Practice Problems

Find the extrema of $(2 x - 1)^{2} + 3$ on the real line and identify both the input and the extreme value.
Find the extrema of $- ∣ x + 1∣ + 3$ , explaining why a derivative is unnecessary for the first conclusion.
Find the largest and smallest values of $sin 2 x + 5$ on the real line. Explain why an extreme value may be attained repeatedly.
Compare the extrema of $x + 1$ on $(- 1, 1)$ and $[- 1, 1]$ .
Explain why $x^{3} + 1$ has a stationary point but no overall maximum or minimum on the real line.
Give the critical points of $1/ (x^{2} + 2)$ , and explain why an input outside a function’s domain cannot be a critical point.
Explain why $e^{x}$ on the real line and $lo g x$ on $(0, \infty)$ do not attain extrema.

Key Takeaways

• An extreme value is an output; its location is an input. • Both the formula and the stated domain determine whether extrema exist. • A local comparison uses nearby values; an absolute comparison uses the whole domain. • Critical points include zero derivatives and undefined derivatives at allowed inputs. • A stationary point must be tested before it is classified.

Previous · Lesson 2

Increasing and Decreasing Functions

Next · Lesson 4

Testing for Local Extrema

The Largest Value Depends on the Allowed Inputs

A Sharp Corner Can Still Be a Minimum

Approaching a Bound Is Not Attaining It

Local Peaks and Valleys

Critical Points Are Candidates

Check Your Understanding

Quiz

Practice Problems

Key Takeaways