Mean Value Theorem Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Mean 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

f

is continuous on

[a, b]

and differentiable on

(a, b)

, then there exists at least one point

c

(a, b)

where

f'(c) = \frac{f(b) - f(a)}{b - a}

If you drive 150 miles in 2 hours, your average speed is 75 mph. The MVT says at some instant during the trip, your speedometer read exactly 75 mph. The instantaneous rate must equal the average rate at least once.

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: On a smooth arc, some interior point has a tangent slope equal to the secant slope across the whole interval.

Common stuck point: The procedure for mean value theorem is the easy part; the trap is skipping the differentiability check. Asking "Is $f$ continuous on $[a,b]$ and differentiable on $(a,b)$ , with a guarantee sought that some $c$ matches the average slope?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is $f$ continuous on $[a,b]$ and differentiable on $(a,b)$ , with a guarantee sought that some $c$ matches the average slope?

Worked Examples

Example 1

easy

Verify the MVT for

f(x) = x^2

[1, 3]

by finding the value of

c

Answer

c = 2 \in (1, 3)

, confirming the MVT.

First step

Check hypotheses:

f

is a polynomial, so continuous on

[1,3]

and differentiable on

(1,3)

. ✓

Full solution

2
Average rate of change: $\frac{f(3)-f(1)}{3-1} = \frac{9-1}{2} = 4$ .
3
Find $c$ : $f'(x) = 2x$ . Set $2c = 4 \Rightarrow c = 2$ .
4
Check: $c = 2 \in (1, 3)$ . ✓

The MVT guarantees a

c

where the instantaneous rate equals the average rate. Here

c=2

is where the tangent to

y = x^2

is parallel to the secant through

(1,1)

and

(3,9)

Example 2

hard

Use the MVT to prove: if

f'(x) = 0

for all

x

(a, b)

, then

f

is constant on

(a, b)

Example 3

medium

Use MVT to prove

|\cos a-\cos b|\le|a-b|

for all real

a,b

Practice Problems

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

Example 1

easy

Find the value of

c

guaranteed by the MVT for

f(x) = x^3

[0, 2]

Example 2

medium

A car travels 120 miles in 2 hours. Explain why the MVT guarantees the car exceeded 60 mph at some instant.

Example 3

easy

Find the average rate of change of

f(x)=x^2

[1,3]

Example 4

easy

Does

f(x)=|x|

satisfy the MVT hypotheses on

[-1,1]

Example 5

easy

For

f(x)=3x+2

[0,5]

, what is

f'(c)

from the MVT?

Example 6

easy

f(x)=x^2

continuous on

[1,4]

and differentiable on

(1,4)

Example 7

easy

What does the MVT conclude geometrically?

Example 8

easy

Driving

120

miles in

2

hours, the MVT guarantees what instantaneous speed occurred?

Example 9

easy

Compute

\frac{f(b)-f(a)}{b-a}

for

f(x)=x^3

[0,2]

Example 10

easy

Rolle's theorem is the MVT under what extra condition?

Example 11

medium

Find all

c

guaranteed by the MVT for

f(x)=x^2

[1,3]

Example 12

medium

Find

c

from the MVT for

f(x)=x^3

[0,2]

Example 13

medium

For

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

[1,2]

, find the MVT point

c

Example 14

medium

Verify the MVT hypotheses fail for

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

[-1,1]

Example 15

medium

Use the MVT to show

|\sin a-\sin b|\le|a-b|

Example 16

medium

f'(x)=0

for all

x

(a,b)

, what does the MVT imply about

f

Example 17

medium

Two functions satisfy

f'(x)=g'(x)

on an interval. What does the MVT-based corollary say?

Example 18

medium

For

f(x)=\sqrt{x}

[0,4]

, find the MVT point

c

Example 19

medium

For

f(x)=x^2-x

[0,2]

, find the MVT point

c

Example 20

challenge

Prove that

e^x\ge1+x

for all

x

using the MVT.

Example 21

challenge

Show that a cubic

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

has its MVT point off-center on

[-2,2]

Example 22

challenge

f

is differentiable with

f(0)=0

and

f'(x)\le2

for all

x

, bound

f(3)

Example 23

easy

Find the average rate of change of

f(x)=x^2

[2,5]

Example 24

easy

For

f(x)=x^2

[1,4]

, find the

c

guaranteed by MVT.

Example 25

easy

For

f(x)=2x+1

[0,3]

, what does the MVT give for

f'(c)

Example 26

easy

Why does

f(x)=\lfloor x\rfloor

fail to satisfy MVT on

[0,2]

Example 27

medium

Find all

c

guaranteed by the MVT for

f(x)=x^2-4x

[0,4]

Example 28

medium

Find

c

from MVT for

f(x)=\sin x

[0,\pi]

Example 29

medium

For

f(x)=\ln x

[1,e]

, find

c

from the MVT.

Example 30

medium

For

f(x)=x^3-3x

[-2,2]

, find the MVT point(s)

c

Example 31

medium

Why does

f(x)=x^{2/3}

fail MVT hypotheses on

[-1,1]

Example 32

medium

A function satisfies

f(2)=5

and

f'(x)\le 3

for all

x

. Find an upper bound for

f(6)

Example 33

medium

A runner covers

400

m in

50

s. Use MVT to justify that at some instant the runner's speed was exactly

8

m/s.

Example 34

medium

For

f(x)=e^x

[0,1]

, find the value of

c

from the MVT.

Example 35

medium

For

f(x)=x^2+x

[1,4]

, find

c

from the MVT.

Example 36

hard

Prove using MVT: if

f'(x)>0

(a,b)

, then

f

is strictly increasing on

(a,b)

Example 37

hard

Suppose

f

is differentiable with

f(1)=2

and

f'(x)\ge 4

[1,5]

. Find a lower bound for

f(5)

Example 38

hard

Use MVT to prove: if

f'

is bounded by

M

in absolute value on an interval, then

|f(x)-f(y)|\le M|x-y|

for any

x,y

in the interval.

Example 39

hard

For

f(x)=\sqrt{x+1}

[0,3]

, find

c

from MVT.

Example 40

medium

For

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

[1,3]

, find

c

from the MVT.

Example 41

hard

Suppose

f

is differentiable on

\mathbb R

f(0)=0

, and

f'(x)\le 1

. Use MVT to bound

f(x)

for

x>0

Example 42

challenge

Two differentiable functions

f

and

g

satisfy

f(0)=g(0)

and

f'(x)<g'(x)

for all

x>0

. Prove

f(x)<g(x)

for

x>0

Example 43

challenge

Show that the equation

x^3+x-1=0

has exactly one real root using MVT (or its monotonicity corollary).

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Related Concepts

Derivative Limit Intermediate Value Theorem Curve Sketching

Background Knowledge

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

derivativelimitintermediate value theorem