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[music]. The mean value theorem has some other

applications where it connects derivative information back to information about the

original function. Well, here's a theorem.

Suppose the derivative of some function. So I'm assuming it's a differential

function, is positive, it's greater than zero on some open interval.

Then the function is an increasing function, meaning that it sends, bigger

inputs to bigger outputs. You probably already believe that

statement. But the mean value theorem let's us verify

that it's actually true. Okay, so let's prove, this result.

I'm going to pick, two points a and b, but I want to make sure that I pick them in

order. So I'm going to pick b bigger than a so I

can talk about the interval a b. All right.

So I'll pick my two points. Now I'm going to apply the mean value

theorem to the open interval a,b. And that's OK because I'm going to pick a

and b to be in this open interval and that's mean that f being differentiable

satisfies the conditions of the mean value theorem.

The continuous on the closed interval a, b and it will be differnetial on the open

interval a, b. Okay.

So the mean value theorem then tells me to look at f of v minus f of a over b minus

a, and the conclusion is that this is the derivative of f at some mystery point in

between a and b. Now the good news is I don't know much

about the derivative but I know the derivative is positive.

So I know that f of b minus f of a over b minus a is positive.

The other thing that I know is that b is greater than a and that means the

denominator here b minus a is also positive.

So I've got mystery number divided by positive is positive.

That means this numerator must also be positive.

So f of b minus f of a is positive. Now, if I add f of a to both sides, I get

that f of b is bigger than f of a. If f sends bigger inputs to bigger outputs

then f is increasing. Increasing means that a bigger input is

sent to a bigger output. So if b is greater than a then f of b is

bigger than f of a and this argument works for any a and b In the open interval on

which I know the derivative is positive. So I'm going to conclude that f is an

increasing function, which is exactly what I wanted to show.

So if the derivative's positive on a whole interval, then the function's increasing

on that interval. But it's important to point out that I

needed to know the sign, the s i g n of the derivative on that entire interval.

The mean value theorem ends of just picking out a single point, that point c,

where it'll examine the derivative. But in order to cover all my bases right

because I don't know what point c the mean value theorem might make me look at .

I've got to control the sign the S I G N on the derivative on the entire interval.

Alright, now I can play the same game when the sign of the derivative is negative.

So, if I know the sign the SIG derivative is negative on some open interval then I

include that F is a decreasing function. And, again this is just an application of

the mean value theorem. Let's see the proof.

So, I'm going to start out by again picking two points a,b in my opening

interval and I'll again pick them so that b is greater than a so I can talk about

the interval a,b. And then applying the mean value theorem,

I should look at, f of b minus, f of a over b minus a.

And the mean value theorem promises me that this is the derivative at some point,

and I don't know what c is but it's the derivative at some point.

But I know that the derivative at any point Is negative so this must be

negative. Now b minus a is positive because b's

bigger than a. So I've got a number I don't know divided

by a positive number is negative. That means that f of b minus f of a must

be negative. Then I add f of a to both sides and I can

conclude that f of b is less than f of a. In other words, f sends bigger inputs to

smaller output. Since f sends bigger inputs to smaller

outputs, f is a decreasing function. Once again we're seeing that the mean

value theorem lets me relate information about the derivative, in this case the

fact that the sign of the derivative is negative.

Back to information about the original function, in this case the function's

decreasing. Alright, well lets summarize what we've

learned thus far. So here are some applications of the mean

value theorem. If f is differentiable on some open

interval and depending as to what happens then we know something about f on that

interval. So if the derivative is identically zero,

if the derivative is equal to zero no matter what I plug in, then f is constant

on that interval. If the derivative is positive no matter

what I plug in, then f is increasing on that interval.

And, if the derivative is negative, then f is decreasing on that interval.