Local vs Global Behavior Examples: 46 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Local vs Global Behavior.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Local behavior describes a function's properties near a specific point; global behavior describes its overall properties across the entire domain or as inputs grow without bound.

Local is "zoom in on one spot"; global is "zoom out to see the whole picture." Near $x = 0$ , $\sin(x) \approx x$ (local linear approximation), but globally it oscillates forever.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Local behavior is what a function does near one point; global behavior is its overall shape across the whole domain.

Common stuck point: The procedure for local vs global behavior is the easy part; the trap is generalizing a local fact to the whole function. Asking "Is the question about the function right around one point, or about its behavior across the whole domain?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the question about the function right around one point, or about its behavior across the whole domain?

Worked Examples

Example 1

easy

For

f(x)=x^3-3x

, describe: (a) local behavior near

x=0

using the linear approximation, and (b) global behavior as

x\to\pm\infty

Answer

Local near

x=0

f(x)\approx-3x

(decreasing line); Global:

f(x)\to\pm\infty

with

x^3

dominance

First step

Local at

x=0

f'(x)=3x^2-3

f'(0)=-3

. Linear approximation:

f(x)\approx f(0)+f'(0)\cdot x = 0+(-3)x=-3x

near

x=0

. Locally the function looks like a line with slope

-3

Full solution

2
Global behavior: for large $|x|$ , the $x^3$ term dominates. $f(x)\to+\infty$ as $x\to+\infty$ ; $f(x)\to-\infty$ as $x\to-\infty$ .
3
Contrast: locally (near $x=0$ ) the function decreases; globally it grows without bound in both directions.

Local behavior captures the function's character near a specific point, often described by a tangent line. Global behavior describes what happens as

x

goes to infinity, usually governed by the highest-degree term.

Example 2

medium

For

g(x) = \dfrac{\sin(x)}{x}

(with

g(0)=1

), describe local behavior near

x=0

and global behavior for large

|x|

Example 3

medium

For

f(x) = \dfrac{1}{1 + x^2}

, describe local behavior near

x = 0

and global behavior as

x \to \pm\infty

Example 4

medium

For

f(x) = \dfrac{x}{x^2 + 4}

, find the global max for

x > 0

Example 5

hard

[1, 4]

, find the global max and min of

f(x) = x + \dfrac{4}{x}

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

For

p(x) = x^4 - 100x^2 + 1

, which term dominates (a) near

x=0

and (b) for

|x| = 1000

Example 2

hard

The function

f(x) = e^{-x^2}

appears bell-shaped. Describe its local behavior at

x=0

(using the second-order Taylor expansion) and global behavior as

x\to\pm\infty

Example 3

easy

Near

x = 0

\sin(x) \approx x

. Is the approximation

\sin(x) \approx x

a statement about local or global behavior?

Example 4

easy

x \to +\infty

, what happens to

f(x) = x^2

? Is this a local or global question?

Example 5

easy

f(x) = -x^2 + 10

has a peak at

x = 0

with value 10. Is this peak a local maximum, a global maximum, or both?

Example 6

easy

A function has a small bump up to height 4 near

x = 1

, but reaches height 20 near

x = 9

. Is the height-4 point a local or global maximum?

Example 7

easy

Near

x = 5

f(x) = -x^2 + 10

is decreasing. Can you conclude it decreases everywhere?

Example 8

easy

Zooming way in on the graph of

f(x) = x^2

x = 2

, it looks almost like a straight line. What is this called?

Example 9

easy

Which describes GLOBAL behavior: 'the slope at

x=2

is 4' or 'the function tends to

+\infty

x\to\infty

Example 10

easy

Does knowing

f(0) = 3

tell you the global maximum of

f

Example 11

medium

For

f(x) = x^3 - 3x

, find all local extrema (slope is

3x^2 - 3

) and state whether

f

has a global maximum.

Example 12

medium

f(x) = x^4 - 4x^2

(slope

4x^3 - 8x

). It has critical points at

x = 0, \pm\sqrt{2}

. Given

f(0)=0

and

f(\pm\sqrt2) = -4

, identify the global minimum.

Example 13

medium

Near

x = 0

f(x) = e^x \approx 1 + x

. Use the local approximation to estimate

e^{0.1}

, and explain why it fails for

x = 5

Example 14

medium

A model fits data well for

0 \le x \le 10

but predicts a negative population at

x = 50

. Is the failure a local or global (extrapolation) problem, and what is the lesson?

Example 15

medium

f(x) = \frac{1}{x}

has a vertical asymptote at

x = 0

and a horizontal asymptote

y = 0

. Classify each asymptote statement as local or global behavior.

Example 16

medium

Two functions agree to a high order near

x=0

f(x) = x

and

g(x) = \sin x

. State one local similarity and one global difference.

Example 17

medium

On the interval

[1, 4]

, find the global maximum of

f(x) = -(x-2)^2 + 5

, checking the vertex and the endpoints.

Example 18

challenge

f(x) = x^3 - 3x

on the closed interval

[-2, 2]

. Find the global maximum and minimum, comparing critical points and endpoints.

Example 19

challenge

A function is locally increasing at every point in

(0, 10)

yet you are told it is NOT globally increasing on the reals. Construct a simple example and explain the apparent paradox.

Example 20

challenge

Explain why a polynomial of odd degree must have a global behavior that runs from

-\infty

+\infty

(or reverse), but can still have local maxima that are not global.

Example 21

medium

f(x) = \frac{x}{x^2 + 1}

has a local maximum value of

0.5

x = 1

. As

x \to \infty

f(x) \to 0

. Is the local maximum also the global maximum for

x > 0

Example 22

medium

f(x) = \sin x

has period

2\pi

. Describe its local behavior near

x = 0

versus its global behavior over all reals.

Example 23

easy

For

f(x) = x^2

, describe the global behavior as

x \to \pm \infty

Example 24

easy

Near

x = 0

\cos(x) \approx 1 - \tfrac{x^2}{2}

. Is this a local or global statement?

Example 25

easy

On the closed interval

[0, 4]

f(x) = x

has a global maximum of what value?

Example 26

easy

For

f(x) = \sin(x)

, what are the global maximum and minimum values?

Example 27

easy

Does

f(x) = e^x

have a global maximum on

\mathbb{R}

Example 28

medium

For

f(x) = x^3 - 12x

, find all local extrema. (Slope is

3x^2 - 12

Example 29

medium

[-3, 3]

, find the global max and min of

f(x) = x^3 - 12x

Example 30

medium

For

f(x) = x^4 - 8x^2 + 5

, find the local minima. (Slope

4x^3 - 16x

Example 31

medium

Use the local approximation

\ln(1 + x) \approx x

near

x = 0

to estimate

\ln(1.05)

Example 32

medium

For

f(x) = x^3

, does the local behavior at

x = 0

(where

f'(0) = 0

) tell you the function has a local extremum there?

Example 33

medium

A polynomial has odd degree with positive leading coefficient. Describe its global end behavior.

Example 34

medium

[0, 2\pi]

, find the global maximum and minimum of

\sin x + \cos x

Example 35

hard

For

f(x) = xe^{-x}

[0, \infty)

, find the global maximum.

Example 36

hard

A function

f

satisfies

f(0) = 10

and has local minima at

x = 1

and

x = 5

with values

f(1) = 2

and

f(5) = -3

. What is the global minimum on

\mathbb{R}

, assuming end behavior

f(x) \to +\infty

x \to \pm \infty

Example 37

hard

For

f(x) = \dfrac{x^2 - 1}{x^2 + 1}

, find the horizontal asymptote and global range.

Example 38

hard

Construct a function whose local maximum value (at one point) is smaller than its value somewhere else.

Example 39

hard

For

f(x) = \ln(1 + x)

near

x = 0

: give the local quadratic approximation.

Example 40

hard

Show that

f(x) = x + \sin x

is globally increasing on

\mathbb{R}

Example 41

challenge

A continuous function

f

[0, 10]

is locally increasing at every point of the open interval

(0, 10)

. Must

f

be globally increasing on

[0, 10]

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Related Concepts

Function

Background Knowledge

These ideas may be useful before you work through the harder examples.

function definition