# 4.4 The mean value theorem

 Page 1 / 7
• Explain the meaning of Rolle’s theorem.
• Describe the significance of the Mean Value Theorem.
• State three important consequences of the Mean Value Theorem.

The Mean Value Theorem is one of the most important theorems in calculus. We look at some of its implications at the end of this section. First, let’s start with a special case of the Mean Value Theorem, called Rolle’s theorem.

## Rolle’s theorem

Informally, Rolle’s theorem states that if the outputs of a differentiable function $f$ are equal at the endpoints of an interval, then there must be an interior point $c$ where $f\prime \left(c\right)=0.$ [link] illustrates this theorem.

## Rolle’s theorem

Let $f$ be a continuous function over the closed interval $\left[a,b\right]$ and differentiable over the open interval $\left(a,b\right)$ such that $f\left(a\right)=f\left(b\right).$ There then exists at least one $c\in \left(a,b\right)$ such that $f\prime \left(c\right)=0.$

## Proof

Let $k=f\left(a\right)=f\left(b\right).$ We consider three cases:

1. $f\left(x\right)=k$ for all $x\in \left(a,b\right).$
2. There exists $x\in \left(a,b\right)$ such that $f\left(x\right)>k.$
3. There exists $x\in \left(a,b\right)$ such that $f\left(x\right)

Case 1: If $f\left(x\right)=0$ for all $x\in \left(a,b\right),$ then $f\prime \left(x\right)=0$ for all $x\in \left(a,b\right).$

Case 2: Since $f$ is a continuous function over the closed, bounded interval $\left[a,b\right],$ by the extreme value theorem, it has an absolute maximum. Also, since there is a point $x\in \left(a,b\right)$ such that $f\left(x\right)>k,$ the absolute maximum is greater than $k.$ Therefore, the absolute maximum does not occur at either endpoint. As a result, the absolute maximum must occur at an interior point $c\in \left(a,b\right).$ Because $f$ has a maximum at an interior point $c,$ and $f$ is differentiable at $c,$ by Fermat’s theorem, $f\prime \left(c\right)=0.$

Case 3: The case when there exists a point $x\in \left(a,b\right)$ such that $f\left(x\right) is analogous to case 2, with maximum replaced by minimum.

An important point about Rolle’s theorem is that the differentiability of the function $f$ is critical. If $f$ is not differentiable, even at a single point, the result may not hold. For example, the function $f\left(x\right)=|x|-1$ is continuous over $\left[-1,1\right]$ and $f\left(-1\right)=0=f\left(1\right),$ but $f\prime \left(c\right)\ne 0$ for any $c\in \left(-1,1\right)$ as shown in the following figure.

Let’s now consider functions that satisfy the conditions of Rolle’s theorem and calculate explicitly the points $c$ where $f\prime \left(c\right)=0.$

## Using rolle’s theorem

For each of the following functions, verify that the function satisfies the criteria stated in Rolle’s theorem and find all values $c$ in the given interval where $f\prime \left(c\right)=0.$

1. $f\left(x\right)={x}^{2}+2x$ over $\left[-2,0\right]$
2. $f\left(x\right)={x}^{3}-4x$ over $\left[-2,2\right]$
1. Since $f$ is a polynomial, it is continuous and differentiable everywhere. In addition, $f\left(-2\right)=0=f\left(0\right).$ Therefore, $f$ satisfies the criteria of Rolle’s theorem. We conclude that there exists at least one value $c\in \left(-2,0\right)$ such that $f\prime \left(c\right)=0.$ Since $f\prime \left(x\right)=2x+2=2\left(x+1\right),$ we see that $f\prime \left(c\right)=2\left(c+1\right)=0$ implies $c=-1$ as shown in the following graph.
2. As in part a. $f$ is a polynomial and therefore is continuous and differentiable everywhere. Also, $f\left(-2\right)=0=f\left(2\right).$ That said, $f$ satisfies the criteria of Rolle’s theorem. Differentiating, we find that $f\prime \left(x\right)=3{x}^{2}-4.$ Therefore, $f\prime \left(c\right)=0$ when $x=\text{±}\frac{2}{\sqrt{3}}.$ Both points are in the interval $\left[-2,2\right],$ and, therefore, both points satisfy the conclusion of Rolle’s theorem as shown in the following graph.

#### Questions & Answers

what is the power rule
Vanessa Reply
how do i deal with infinity in limits?
Itumeleng Reply
Add the functions f(x)=7x-x g(x)=5-x
Julius Reply
f(x)=7x-x g(x)=5-x
Awon
5x-5
Verna
what is domain
Cabdalla Reply
difference btwn domain co- domain and range
Cabdalla
x
Verna
The set of inputs of a function. x goes in the function, y comes out.
Verna
where u from verna
Arfan
If you differentiate then answer is not x
Raymond
domain is the set of values of independent variable and the range is the corresponding set of values of dependent variable
Champro
what is functions
mahin Reply
give different types of functions.
Paul
how would u find slope of tangent line to its inverse function, if the equation is x^5+3x^3-4x-8 at the point(-8,1)
riyad Reply
pls solve it i Want to see the answer
Sodiq
ok
Friendz
differentiate each term
Friendz
why do we need to study functions?
abigail Reply
to understand how to model one variable as a direct relationship to another variable
Andrew
integrate the root of 1+x²
Rodgers Reply
use the substitution t=1+x. dt=dx √(1+x)dx = √tdt = t^1/2 dt integral is then = t^(1/2 + 1) / (1/2 + 1) + C = (2/3) t^(3/2) + C substitute back t=1+x = (2/3) (1+x)^(3/2) + C
navin
find the nth differential coefficient of cosx.cos2x.cos3x
Sudhanayaki Reply
determine the inverse(one-to-one function) of f(x)=x(cube)+4 and draw the graph if the function and its inverse
Crystal Reply
f(x) = x^3 + 4, to find inverse switch x and you and isolate y: x = y^3 + 4 x -4 = y^3 (x-4)^1/3 = y = f^-1(x)
Andrew
in the example exercise how does it go from -4 +- squareroot(8)/-4 to -4 +- 2squareroot(2)/-4 what is the process of pulling out the factor like that?
Robert Reply
can you please post the question again here so I can see what your talking about
Andrew
√(8) =√(4x2) =√4 x √2 2 √2 hope this helps. from the surds theory a^c x b^c = (ab)^c
Barnabas
564356
Myong
can you determine whether f(x)=x(cube) +4 is a one to one function
Crystal
one to one means that every input has a single output, and not multiple outputs. whenever the highest power of a given polynomial is odd then that function is said to be odd. a big help to help you understand this concept would be to graph the function and see visually what's going on.
Andrew
one to one means that every input has a single output, and not multiple outputs. whenever the highest power of a given polynomial is odd then that function is said to be odd. a big help to help you understand this concept would be to graph the function and see visually what's going on.
Andrew
can you show the steps from going from 3/(x-2)= y to x= 3/y +2 I'm confused as to how y ends up as the divisor
Robert Reply
step 1: take reciprocal of both sides (x-2)/3 = 1/y step 2: multiply both sides by 3 x-2 = 3/y step 3: add 2 to both sides x = 3/y + 2 ps nice farcry 3 background!
Andrew
first you cross multiply and get y(x-2)=3 then apply distribution and the left side of the equation such as yx-2y=3 then you add 2y in both sides of the equation and get yx=3+2y and last divide both sides of the equation by y and you get x=3/y+2
Ioana
Multiply both sides by (x-2) to get 3=y(x-2) Then you can divide both sides by y (it's just a multiplied term now) to get 3/y = (x-2). Since the parentheses aren't doing anything for the right side, you can drop them, and add the 2 to both sides to get 3/y + 2 = x
Melin
thank you ladies and gentlemen I appreciate the help!
Robert
keep practicing and asking questions, practice makes perfect! and be aware that are often different paths to the same answer, so the more you familiarize yourself with these multiple different approaches, the less confused you'll be.
Andrew
please how do I learn integration
aliyu Reply
they are simply "anti-derivatives". so you should first learn how to take derivatives of any given function before going into taking integrals of any given function.
Andrew
best way to learn is always to look into a few basic examples of different kinds of functions, and then if you have any further questions, be sure to state specifically which step in the solution you are not understanding.
Andrew
example 1) say f'(x) = x, f(x) = ? well there is a rule called the 'power rule' which states that if f'(x) = x^n, then f(x) = x^(n+1)/(n+1) so in this case, f(x) = x^2/2
Andrew
great noticeable direction
Isaac
limit x tend to infinite xcos(π/2x)*sin(π/4x)
Abhijeet Reply
can you give me a problem for function. a trigonometric one
geovanni Reply

### Read also:

#### Get the best Calculus volume 1 course in your pocket!

Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?

 By By Anindyo Mukhopadhyay By