Main content

### Course: AP®︎/College Calculus AB > Unit 5

Lesson 1: Using the mean value theorem- Mean value theorem
- Mean value theorem example: polynomial
- Mean value theorem example: square root function
- Using the mean value theorem
- Justification with the mean value theorem: table
- Justification with the mean value theorem: equation
- Establishing differentiability for MVT
- Justification with the mean value theorem
- Mean value theorem application
- Mean value theorem review

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Mean value theorem example: polynomial

Sal finds the number that satisfies the Mean value theorem for f(x)=x²-6x+8 over the interval [2,5]. Created by Sal Khan.

## Want to join the conversation?

- How did he get the vertex at 3?(9 votes)
- It appears he was relying on the symmetry of a parabola, as he says it's halfway between the x-intercepts. There are other ways to get this however. Note that f'(x) = 2x - 6. Setting that equal to zero tells us the critical point is at x = 3, and in the case of a parabola, there is only one critical point, at the vertex.(36 votes)

- Would the average rate of change over an interval be the same as the value guaranteed by the Mean Value Theorem over the same interval?(2 votes)
- The Mean Value Theorem doesn't guarantee any particular value or set of values. Rather, it states that for any closed interval over which a function is continuous, there exists some x within that interval at which the slope of the tangent equals the slope of the secant line defined by the interval endpoints.(10 votes)

- how does he find that f'(x) = 2x-6(1 vote)
`f(x) = x² - 6x +8`

`f'(x) = 2x - 6`

If you've watched the previous videos on taking derivatives, you should know the power rule which states:

if`f(x) = x^n`

, then`f'(x) = n*x^(n-1)`

.

You should also know the rule if`f(x) = g(x) + h(x)`

then`f'(x) = g'(x) + h'(x)`

(12 votes)

- At0:55, why does Sal convert from brackets to parenthesis? Don't those represent closed and open intervals?(1 vote)
- That's correct. Recall that the statement of the mean value theorem requires that the function be continuous on the closed interval [a, b], but differentiable only on the open interval (a, b). Sal switches to open intervals when he begins talking about the derivative of the function.(5 votes)

- Why did he include the vertex on the graph about the mean theorem? plz explain this(1 vote)
- Sal did it to help himself graph the equation better.(5 votes)

- So does c represent the instant slope/rate of change/derivative? Or is it an x value were the derivative and secant slope are equal?(2 votes)
- c just represents the point where the instantaneous slope is equal to the average slope of the function in the closed interval.(3 votes)

- I'm just curious, if the equation wasn't able to be factored, would two be the only value then that has a y-value of zero? I just want to make sure for future understanding. Thank you!(2 votes)
- If the equation is impossible/difficult to factor,

we would use the quadratic formula

to get the roots/zeros for the equation (read when y=0.. the x-intercept)(3 votes)

- How to calculate average f' over a period (a,b)? Is this equal to average rate of change over (a,b)? Thanks.(2 votes)
- How is the x-value the same as a c?(1 vote)
- In the function
`f'(x) = 2x - 6`

,`x`

can be any value, and it would output a specific derivative of`f(x)`

.`c`

is the specific value of`x`

at which`f'(x)`

would be`1`

. In other words,`x`

can be any number that would give you a certain`f'(x)`

, but`c`

is the particular value that`x`

can take on to make`f'(x)`

be`1`

.(3 votes)

- I was curious if the Number of points on the closed interval that satisfy the mean value theorem is = to the (number of inflection points) + 1(2 votes)
- Yes, you are correct!(1 vote)

## Video transcript

Let's say I have
some function f of x that is defined as being equal
to x squared minus 6x plus 8 for all x. And what I want to do is
show that for this function we can definitely find
a c in an interval where the derivative at the point c
is equal to the average rate of change over that interval. So let's give ourselves an
interval right over here. Let's say we care about the
interval between 2 and 5. And this function is definitely
continuous over this closed interval, and it's also
differentiable over it. And it just has to
be differentiable over the open interval, but
this is differentiable really for all x. And so let's show
that we can find a c that's inside the
open interval, that's a member of the open
interval, that's in the open interval such
that the derivative at c is equal to the average rate
of change over this interval. Or is equal to the slope of
the secant line between the two endpoints of the interval. So it's equal to f of 5
minus f of 2 over 5 minus 2. And so I encourage you to
pause the video now and try to find a c where
this is actually true. Well to do that, let's just
calculate what this has to be. Then let's just take the
derivative and set them equal and we should be able
to solve for our c. So let's see f of 5 minus f of
2, f of 5 is, let's see, f of 5 is equal to 25 minus 30 plus 8. So that's negative 5
plus 8 is equal to 3. f of 2 is equal to
2 squared minus 12. So it's 4 minus 12 plus 8. That's going to be a 0. So this is equal to 3/3,
which is equal to 1. f prime of c needs
to be equal to 1. And so what is the
derivative of this? Well let's see, f prime of
x is equal to 2x minus 6. And so we need to figure
out at what x value, especially it has to be
in this open interval, at what x value
is it equal to 1? So this needs to be equal to 1. So let's add 6 to both sides. You get 2x is equal to 7. x is equal to 7/2, which is
the same thing as 3 and 1/2. So it's definitely in this
interval right over here. So we've just found
our c is equal to 7/2. And let's just graph
this to really make sure that this makes sense. So this right over
here is our y-axis. And then this right
over here is our x-axis. Looks like all the
action is happening in the first and
fourth quadrants. So that is our x-axis. And let's see, let's say
this is 1, 2, 3, 4, and 5. So we already know that 2
is one of the zeroes here. So we know that our function
if we wanted to graph it, it intersects the
x-axis right over here. And you can factor this out
as x minus 2 times x minus 4. So the other place where
our function hits 0 is when x is equal
to 4 right over here. Our vertex is going to be right
in between at x is equal to 3. When x is equal to 3, let's
see, 9 minus 18 is negative 9 plus 8, so negative 1. So you have the point
3, negative 1 on it. And so that's enough
for us to graph. And we also know
at 5, we're at 3. So 1, 2, 3. So at 5, we are right over here. So over the interval
that we care about, our graph looks
something like this. So that's the interval
that we care about. And we're saying
that we were looking for a c whose slope is
the same as the slope of the secant line,
same as the slope of the line between
these two points. And if I were to just visually
look at it, I'd say well yeah, it looks like right around
there just based on my drawing, the slope of the tangent line
looks like it's parallel. It looks like it
has the same slope, looks like the tangent line is
parallel to the secant line. And that looks like it's
right at 3 and 1/2 or 7/2. So it makes sense. So this right over here is
our c. c is equal to 7/2.