This video is about monotonicity of functions.
When is a function increasing or decreasing,
and what does this have to do with derivatives?
This will be our third application of the Mean Value Theorem.
Before I begin, an important clarification.
Saying that a function is increasing is NOT the same thing
as saying it has positive derivative.
These two concepts are related, but they are NOT THE SAME.
Positive derivative is NOT the definition of a function increasing.
Here is why.
This is an example of a function that is increasing everywhere,
but it has derivative zero at a POINT.
This graph represents another function that is also increasing everywhere,
but it has a corner at a point, so it is not differentiable at one point.
It doesn't have a derivative there.
This is a function that is increasing everywhere,
and yet, at infinitely many points, it is not even continuous, let alone differentiable.
So, a function can be increasing without having positive derivative.
We should then define "increasing function" formally.
Here is the definition.
I begin with a function defined on an interval I.
I want to say what it means to be increasing on I.
It is always "a function is increasing ON A DOMAIN."
That means this condition.
That, for every two points x_1 and x_2 in I,
if x_1 is less than x_2, THEN f(x_1) is less than f(x_2).
So that means, if I take a bigger input, then it produces a bigger output.
That is going to mean that the graph is always going up.
I am using this definition, where I say that f(x_1) is LESS than f(x_2),
but there is a related concept, which I am going to call NON-DECREASING,
in which case I say that x_1 < x_2 implies
that f(x_1) is less than or equal to f(x_2).
So, increasing just means the graph always "goes up."
And non-decreasing means that the graph always goes up or stays constant.
Of course, there are also the notions of decreasing, and non-increasing,
which will be similar.
I assume you can write the definitions. Just reverse the second inequality.
Unfortunately, this notation is not universal.
While some books use this convention and call these two things
"increasing" and "non-decreasing,"
other books use "strictly increasing" and "increasing" instead,
which is unfortunate, because it means "increasing"
could mean either one of the two,
depending on which math book or math paper you are reading.
There is no way around it. Both are used.
So, if you are reading a paper or a book, make sure you know
which definition it is using. And if you are doing math with somebody else,
agree on a definition before starting.
So, that is the definition.
The three graphs of functions I looked at before satisfy this property.
They are increasing. This is an increasing function.
This is an increasing function.
And this also satisfies the definition, so it is an increasing function.
Okay! Now that we know what increasing means,
what is the relation between increasing and positive derivative?
Here is the theorem.
Let's say I begin with a function f, defined on an open interval (a, b).
IF I know that the derivative of the function is positive,
THEN I can conclude that the function is increasing on that interval.
That is the theorem. Positive derivative implies increasing function.
But, a function could be increasing without having positive derivative.
That is the theorem for an open interval.
There is a companion to this one.
If instead, I am looking at a function defined
on a closed and bounded interval [a, b],
if I want to conclude that the function is increasing on [a, b],
then I only need to require that the derivative is positive
on the OPEN interval – on the interior.
But, on the endpoints, I must require continuity.
So, positive derivative on the interior and continuity everywhere,
implies that the function is increasing on that whole interval.
These two theorems need proofs. I invite you to write them.
A couple of videos ago, a wrote a proof for a similar result,
that if the derivative of a function is zero, then the function must be constant.
That was a direct application of the Mean Value Theorem.
This works in exactly the same way. So I invite you to rewatch that video –
– I will add a link in the description –
– and try to imitate the proof. You only have to modify it slightly
to prove these two theorems.
That is a good exercise to make sure you understand the Mean Value Theorem,
and how to write these kinds of proofs.
So, after seeing the theorems and their proofs,
here is a summary of what we know so far.
On an open interval, IF the derivative is zero,
THEN the function must be constant.
IF the derivative is positive, THEN the function must be increasing.
IF the derivative is negative, THEN the function must be decreasing.
I have not written the theorem for this last case,
but it works exactly the same as the others.
But, if I just have a point, where the derivative is zero or does not exist,
then in principle, if could be anything.
It could be part of a bigger interval where the function is increasing,
or we may have a local extremum at that point, or something else.
In the next video, I will show with an example how to use these theorems
to determine the intervals where a certain function
is increasing or decreasing.