Intermediate Value Theorem Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Intermediate Value Theorem.

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

Concept Recap

If $f$ is continuous on the closed interval $[a, b]$ and $N$ is any value between $f(a)$ and $f(b)$ , then there exists at least one $c$ in $(a, b)$ such that $f(c) = N$ .

A continuous function can't skip values. If you start below a line and end above it, you must cross it somewhere. It's like driving from sea level to a mountaintop—you pass through every elevation in between.

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: If a continuous function starts below a level and ends above it, it must hit that level somewhere between.

Common stuck point: The procedure for intermediate value theorem is the easy part; the trap is applying IVT without checking continuity. Asking "Is the function continuous on a closed interval, and am I asked to show a value between the endpoints is attained somewhere inside?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the function continuous on a closed interval, and am I asked to show a value between the endpoints is attained somewhere inside?

Worked Examples

Example 1

easy

Show that

f(x) = x^3 - x - 1

has a root in the interval

(1, 2)

Answer

By IVT,

f

has at least one root in

(1, 2)

First step

f

is a polynomial, hence continuous on

[1, 2]

Full solution

2
$f(1) = 1 - 1 - 1 = -1 < 0$ .
3
$f(2) = 8 - 2 - 1 = 5 > 0$ .
4
Since $f$ is continuous, $f(1) < 0 < f(2)$ , by the IVT there exists $c \in (1,2)$ with $f(c) = 0$ .

The IVT requires continuity and a sign change. Polynomials are continuous everywhere, so verifying the sign change at the endpoints is sufficient.

Example 2

medium

Use the IVT to show that

\cos x = x

has a solution in

(0, 1)

Example 3

easy

Show

g(x)=x^3+2x-5

has a root in

(1,2)

Example 4

medium

Show

\ln x = 1-x

has a solution in

(1,e)

Example 5

medium

Show the equation

x = \cos(x)

has a solution in

[0,1]

Example 6

hard

Show that every continuous function

f:[0,1] \to [0,1]

has a fixed point.

Practice Problems

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

Example 1

easy

Show that

h(x) = x^2 - 2

has a root in

(1, 2)

Example 2

medium

Can IVT be applied to

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

[-1, 1]

to conclude it attains the value 0? Explain.

Example 3

easy

Does

f(x) = x^2

take the value 2 somewhere on

[0, 2]

? (It is continuous.)

Example 4

easy

Given

f

continuous with

f(1) = -3

and

f(4) = 5

, must

f

have a root in

(1,4)

Example 5

easy

State the two hypotheses required to apply the IVT on

[a,b]

Example 6

easy

Can IVT be applied to

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

[-1, 1]

Example 7

easy

A continuous function has

f(0) = 1

and

f(3) = 7

. Is there a

c

with

f(c) = 4

Example 8

easy

Does

f(x) = \cos x

equal 0 somewhere on

[0, 2]

Example 9

easy

Why must

f

be continuous on the CLOSED interval

[a,b]

for IVT?

Example 10

easy

f

is continuous and

f(2) = f(5) = 3

, does IVT guarantee

f

takes the value 10 on

[2,5]

Example 11

medium

Show

f(x) = x^3 + x - 1

has a root in

(0, 1)

Example 12

medium

Show

\cos x = x

has a solution in

(0, 1)

Example 13

medium

Show

f(x) = x^5 - 2x + 3

has a real root.

Example 14

medium

A continuous temperature is

-2^\circ

at 6am and

10^\circ

at noon. Was it exactly

0^\circ

at some time?

Example 15

medium

Show

f(x) = e^x + x = 2

has exactly one solution, locating an interval.

Example 16

medium

A continuous

f

maps

[0,1]

into

[0,1]

. Show

f

has a fixed point (

f(c)=c

Example 17

medium

Show

\tan x = x

has a solution in

\left(\pi, \frac{3\pi}{2}\right)

... wait, why must we avoid

\frac{\pi}{2}

intervals?

Example 18

medium

Given continuous

f

with

f(1)=2, f(2)=-1, f(3)=4

, what is the minimum number of roots IVT guarantees on

[1,3]

Example 19

medium

Prove every odd-degree polynomial

p(x)

has at least one real root.

Example 20

challenge

A hiker climbs a trail 8am-2pm, descends the same trail next day 8am-2pm. Prove there is a clock time at the same spot both days.

Example 21

challenge

Show

f(x) = x^3 - 3x + 1

has three real roots by locating sign changes.

Example 22

challenge

A continuous

f

[0,1]

satisfies

f(0)=f(1)

. Show some

c

has

f(c) = f(c + \tfrac12)

[0,\tfrac12]

Example 23

easy

A continuous function

f

has

f(0)=-4

and

f(3)=6

. Does the IVT guarantee a value

c \in (0,3)

with

f(c)=1

Example 24

easy

Let

f(x)=x^2-5

. Use IVT to show

f

has a root in

(2,3)

Example 25

easy

A continuous function has

f(-1)=2

and

f(2)=2

. Does IVT guarantee

f(c)=0

for some

c \in (-1,2)

Example 26

easy

Can IVT be applied to

f(x)=\lfloor x \rfloor

[0.5,2.5]

? Why or why not?

Example 27

medium

Show that

f(x)=x^4-3x-1

has a root in

(1,2)

Example 28

medium

Use IVT to show

e^x = 3-x

has a solution in

(0,1)

Example 29

medium

A continuous function on

[0,10]

has

f(0)=-7

and

f(10)=15

. List three values IVT guarantees the function attains on

(0,10)

Example 30

medium

Show

\sin x = \frac{x}{2}

has a positive solution.

Example 31

medium

At noon a car is at mile marker 5; at 1pm it is at mile marker 65. Assuming continuous motion, must it have passed mile marker 40 at some time in between?

Example 32

medium

Show that

f(x)=x^3-6x+2

has at least two roots in

[-3,3]

Example 33

medium

A continuous function

f

[1,5]

has

f(1)=3

and

f(5)=8

. Must

f

attain the value

10

on the interval?

Example 34

hard

Show

2^x = 3x

has a solution in

(0,1)

and another in

(3,4)

Example 35

hard

Let

f

be continuous on

[0,1]

with

f(0)=1

and

f(1)=0

. Prove there exists

c

with

f(c)=c

Example 36

hard

Show that the polynomial

p(x)=x^5-4x^3+x-1

has at least three real roots.

Example 37

hard

Use IVT to approximate the root of

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

(1,2)

to one decimal place by bisection.

Example 38

hard

Two continuous functions

f,g:[0,1] \to \mathbb{R}

satisfy

f(0)<g(0)

and

f(1)>g(1)

. Prove they intersect.

Example 39

challenge

Let

f

be continuous on

[0,2]

with

f(0)=f(2)

. Prove there is

c \in [0,1]

with

f(c)=f(c+1)

Example 40

challenge

Show that the equation

x^7 + x^3 - 2x - 5 = 0

has at least one real solution.

Example 41

challenge

A continuous

f:[0,1] \to \mathbb{R}

satisfies

f(0)+f(1)=1

. Show some

c \in [0,1]

has

f(c)=\tfrac12

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Related Concepts

Limit Types of Continuity and Discontinuity Mean Value Theorem

Background Knowledge

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

limitcontinuity types