[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.