Respected Sir, I have a doubt under "Why continuity at the edges is important.". I did not quite understand in the assumption that is considered, on why is it, that the secant line doesn't join to a point in the curve considered? Isn't it supposed to join to some point before the point b in the curve?

you're thinking about a tangent, a secant is simply a line, from point a to point b

What is the Extreme Value Theorem?

It's the fact that a continuous function on a closed interval will achieve a maximum value and a minimum value.

Why is the below worded as it is. Specifically why not just say that f(x) is differentiable over [a, b]? How is the statement I said different different then the statement below? Why does it matter? My theory is that the end of a line can't be differntiable but i don't know. "The precise conditions under which MVT applies are that fff is differentiable over the open interval (a,b) and continuous over the closed interval [a,b]. Since differentiability implies continuity, we can also describe the condition as being differentiable over (a,b) and continuous at x=a and x=b."

Your suspicion is correct. We can't differentiate the function at the end points of a closed interval. See rashaveraka's answer to this question: https://www.khanacademy.org/math/ap-calculus-ab/ab-existence-theorems/ab-mvt/v/mean-value-theorem-1?qa_expand_key=kaencrypted_2edb39dd806f4a066e87def729042ba5_bdc4f8623227f4d08c2ef1e0f0434660194306685ca9cbbb58337efd9174b0146234d247078dc05fb582dcb55d4dd5843fe41cb067f9826a1ffeea2112cf2c88239f510de888397343745ca05bff49bdefed5141cc558eaf3c818bd9f02ca567c70bcc3fc621e988078de6256e9333e17ac868f66fcdf5835e90eec38fc7bf187b9bef4b0bb8d1552bfd891fa34a90a3d2a7288854c26b4532ac73fdf4c4ac3d

Why does the Mean Value Theorem specify differentiability on an open interval (a,b) rather than a half-open interval like [a,b] or (a,b]? What implications would including an endpoint have on the theorem's validity?

We don't consider open intervals because endpoints aren't differentiable. Differentiability requires the derivative to be same on both sides of a point. But for endpoints, we don't have "both sides"

At the top of this article, it says that the Mean Value Theorem is an existence theorem. What's that mean?

it means that its a theorem that guarantees the existence of a certain point

Great article. I understand why sharp turns would ruin the MVT, but how vertical tangent does?

Remember what the Mean Value Theorem's conditions are. It says that a function needs to be *differentiable* on some interval (a,b). Now, if we have a vertical tangent in said interval, as the slope of the tangent is undefined, our interval is no longer differentiable, as we have an undefined slope at a point. Hence, vertical tangents would make MVT invalid on the interval

Main content

Course: Differential Calculus > Unit 5

Lesson 1: Mean value theorem

Establishing differentiability for MVT

Google Classroom

A function must be differentiable for the mean value theorem to apply. Learn why this is so, and how to make sure the theorem can be applied in the context of a problem.

The mean value theorem (MVT) is an existence theorem similar the intermediate and extreme value theorems (IVT and EVT). Our goal is to understand the mean value theorem and know how to apply it.

MVT and its conditions

The mean value theorem guarantees, for a function

f

that's differentiable over an interval from

a

b

, that there exists a number

c

on that interval such that

f^{'} (c)

is equal to the function's average rate of change over the interval.

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

Graphically, the theorem guarantees that an arc between two endpoints has a point at which the tangent to the arc is parallel to the secant through its endpoints.

The precise conditions under which MVT applies are that

f

is differentiable over the open interval

(a, b)

and continuous over the closed interval

[a, b]

. Since differentiability implies continuity, we can also describe the condition as being differentiable over

(a, b)

and continuous at

x = a

and

x = b

Using parameters like

a

and

b

and talking about open and closed intervals is important if we want to be mathematically precise, but these conditions essentially mean this:

For MVT to apply, the function must be differentiable over the relevant interval, and continuous at the interval's edges.

Why differentiability over the interval is important.

To understand why this condition is important, consider function

f

. The function has a sharp turn between

x = a

and

x = b

, so it's not differentiable over

(a, b)

Indeed, the function has only two possible tangent lines, neither of which is parallel to the secant between

x = a

and

x = b

Why continuity at the edges is important.

To understand this, consider function

g

As long as

g

is differentiable over

(a, b)

and continuous at

x = a

and

x = b

, MVT applies.

Now let's change

g

so it's not continuous at

x = b

. In other words, the one-sided limit

lim_{x \to b^{-}} g (x)

remains the same, but the function value changes to something else.

Notice how all of the possible tangent lines on the interval are necessarily increasing, while the secant line is decreasing. So there isn't any tangent line that's parallel to the secant line.

In general, if a function isn't continuous at the edges, the secant line will be disconnected from the tangent lines along the interval.

In Problem set 1 we will analyze the applicability of the mean value theorem to function

h

at different intervals.

Problem 1.A

Does MVT apply to $h$ ‍ over the interval $[- 5, - 1]$ ‍?

Problem 2

The graph of function

f

has a vertical tangent at

x = 2

Does MVT apply to $f$ ‍ over the interval $[- 1, 5]$ ‍?

Want more practice? Try this exercise.

Notice: When MVT doesn't apply, all we can tell is that we aren't certain the conclusion is true. It does not mean that the conclusion isn't true.

In other words, it's possible to have a point where the tangent is parallel to the secant, even when MVT doesn't apply. We just can't be certain about it unless the conditions for MVT have been met.

For example, in the last problem, MVT didn't apply to

f

over the interval

[- 1, 5]

, even though there are actually two points in the interval

[- 1, 5]

where the tangent is parallel to the secant between the endpoints.

Problem 3

This table gives a few values of function

h

$x$ ‍	$3$ ‍	$7$ ‍	$10$ ‍	$11$ ‍
$h (x)$ ‍	$1$ ‍	$5$ ‍	$- 2$ ‍	$- 6$ ‍

James said that since

\frac{h (7) - h (3)}{7 - 3} = 1

there must be a number

c

in the interval

[3, 7]

for which

h^{'} (c) = 1

Which condition makes James's claim true?

Want more practice? Try this exercise.

Common mistake: Not recognizing when the conditions are met

Let's take Problem 3 for example. These are the common ways we would expect the conditions for MVT to look:

$h$ ‍ is differentiable over $(3, 7)$ ‍ and continuous over $[3, 7]$ ‍.
$h$ ‍ is differentiable over $(3, 7)$ ‍ and continuous at $x = 3$ ‍ and $x = 7$ ‍.

However, we will not always be given information about the function this way. For example, if

h

is differentiable over

[3, 7]

, the conditions are met because differentiability implies continuity.

Another example is when

h

is differentiable over a larger interval, for example

(2, 8)

. Even though continuity isn't mentioned, differentiability over

(2, 8)

implies differentiability over

(3, 7)

and continuity over

[3, 7]

Problem 4

f

is a differentiable function.

f (1) = - 2

and

f (5) = 2

Match each conclusion with its appropriate existence theorem.

1

Common mistake: Applying the wrong existence theorem

By now, we are familiar with three different existence theorems: the intermediate value theorem (IVT), the extreme value theorem (EVT), and the mean value theorem (MVT). They have a similar structure but they apply under different conditions and guarantee different kinds of points.

IVT guarantees a point where the function has a certain value between two given values.
EVT guarantees a point where the function obtains a maximum or a minimum value.
MVT guarantees a point where the derivative has a certain value.

Before applying one of the existence theorems, make sure you understand the problem well enough to tell which theorem should be applied.

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Chris Trimble
Posted 9 months ago. Direct link to Chris Trimble's post “Things seem a bit out of ...”
Things seem a bit out of order ... This article refers to the EVT, but the EVT is not introduced until the next lesson.
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(29 votes)
Answer
Anupama
Posted 7 years ago. Direct link to Anupama's post “Respected Sir, I have ...”
Respected Sir,
I have a doubt under "Why continuity at the edges is important.".
I did not quite understand in the assumption that is considered, on why is it, that the secant line doesn't join to a point in the curve considered? Isn't it supposed to join to some point before the point b in the curve?
Button navigates to signup pageButton navigates to signup page
(13 votes)
Answer
- Thorbjørn Frommelt
  Posted 7 years ago. Direct link to Thorbjørn Frommelt's post “you're thinking about a t...”
  you're thinking about a tangent, a secant is simply a line, from point a to point b
  Button navigates to signup page
  (17 votes)
Seth
Posted 5 years ago. Direct link to Seth's post “What is the Extreme Value...”
What is the Extreme Value Theorem?
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(7 votes)
Answer
- kubleeka
  Posted 5 years ago. Direct link to kubleeka's post “It's the fact that a cont...”
  It's the fact that a continuous function on a closed interval will achieve a maximum value and a minimum value.
  Button navigates to signup page
  (11 votes)
JSperling
Posted a year ago. Direct link to JSperling's post “In problem 3, the questio...”
In problem 3, the question "Which condition makes James's claim true?" is a little misleading. It assumes there is something in the information given that makes the claim true, but there isn't enough information to make that determination. A better question would be "Which condition WOULD MAKE James's claim true?"
Button navigates to signup pageComment on JSperling's post “In problem 3, the questio...”
(6 votes)
Answer
Garret Cervantez
Posted 7 years ago. Direct link to Garret Cervantez's post “Why is the below worded a...”
Why is the below worded as it is. Specifically why not just say that f(x) is differentiable over [a, b]? How is the statement I said different different then the statement below? Why does it matter?

My theory is that the end of a line can't be differntiable but i don't know.

"The precise conditions under which MVT applies are that fff is differentiable over the open interval (a,b) and continuous over the closed interval [a,b]. Since differentiability implies continuity, we can also describe the condition as being differentiable over (a,b) and continuous at x=a and x=b."
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- Howard Bradley
  Posted 7 years ago. Direct link to Howard Bradley's post “Your suspicion is correct...”
  Your suspicion is correct. We can't differentiate the function at the end points of a closed interval.
  See rashaveraka's answer to this question:
  https://www.khanacademy.org/math/ap-calculus-ab/ab-existence-theorems/ab-mvt/v/mean-value-theorem-1?qa_expand_key=kaencrypted_2edb39dd806f4a066e87def729042ba5_bdc4f8623227f4d08c2ef1e0f0434660194306685ca9cbbb58337efd9174b0146234d247078dc05fb582dcb55d4dd5843fe41cb067f9826a1ffeea2112cf2c88239f510de888397343745ca05bff49bdefed5141cc558eaf3c818bd9f02ca567c70bcc3fc621e988078de6256e9333e17ac868f66fcdf5835e90eec38fc7bf187b9bef4b0bb8d1552bfd891fa34a90a3d2a7288854c26b4532ac73fdf4c4ac3d
  Comment on Howard Bradley's post “Your suspicion is correct...”
  (5 votes)
2VETURIrithvik
Posted 3 years ago. Direct link to 2VETURIrithvik's post “Wait, I don't remember le...”
Wait, I don't remember learning about IVT and EVT yet. This page said, "By now, we are familiar with three different existence theorems: the intermediate value theorem (IVT), the extreme value theorem (EVT), and the mean value theorem (MVT)."

Does anyone know which unit that is in?
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(6 votes)
Answer
- Ash_001
  Posted 3 years ago. Direct link to Ash_001's post “IVT was in "Limits and Co...”
  IVT was in "Limits and Continuity" and EVT was in "Applying Derivatives to analyze functions."
  Comment on Ash_001's post “IVT was in "Limits and Co...”
  (0 votes)
aphinoo.damian
Posted 5 months ago. Direct link to aphinoo.damian's post “Why does the Mean Value T...”
Why does the Mean Value Theorem specify differentiability on an open interval (a,b) rather than a half-open interval like [a,b] or (a,b]? What implications would including an endpoint have on the theorem's validity?
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Venkata
  Posted 5 months ago. Direct link to Venkata's post “We don't consider open in...”
  We don't consider open intervals because endpoints aren't differentiable. Differentiability requires the derivative to be same on both sides of a point. But for endpoints, we don't have "both sides"
  Button navigates to signup page
  (2 votes)
Gavin Anderson
Posted a month ago. Direct link to Gavin Anderson's post “At the top of this articl...”
At the top of this article, it says that the Mean Value Theorem is an existence theorem. What's that mean?
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(1 vote)
Answer
- Forever Learner
  Posted a month ago. Direct link to Forever Learner's post “it means that its a theor...”
  it means that its a theorem that guarantees the existence of a certain point
  Comment on Forever Learner's post “it means that its a theor...”
  (3 votes)
Mohamed Ibrahim
Posted a year ago. Direct link to Mohamed Ibrahim's post “Great article. I understa...”
Great article. I understand why sharp turns would ruin the MVT, but how vertical tangent does?
Button navigates to signup pageComment on Mohamed Ibrahim's post “Great article. I understa...”
(1 vote)
Answer
- Venkata
  Posted a year ago. Direct link to Venkata's post “Remember what the Mean Va...”
  Remember what the Mean Value Theorem's conditions are. It says that a function needs to be differentiable on some interval (a,b). Now, if we have a vertical tangent in said interval, as the slope of the tangent is undefined, our interval is no longer differentiable, as we have an undefined slope at a point. Hence, vertical tangents would make MVT invalid on the interval
  Comment on Venkata's post “Remember what the Mean Va...”
  (2 votes)
Connie Williamson
Posted 4 years ago. Direct link to Connie Williamson's post “In the first MVT video Sa...”
In the first MVT video Sal says that the rules for MVT are that it is continuous for the entire interval [a, b] but only has to be differentiable for the interval inside (a,b).
Then, in the first section of questions on this article, it says "hhh is differentiable over the entire interval between x=-3x=−3x, equals, minus, 3 and x=2x=2x, equals, 2, but it isn't continuous at x=-3x=−3x, equals, minus, 3.
The answer is no."

Which one is right?
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(1 vote)
Answer