What I have here in yellow is the graph of y equals f of x.
Then here in this mauve color I've
graphed y is equal to the derivative of f
is f prime of x.
And then here in blue, I've graphed
y is equal to the second derivative of our function.
So this is the derivative of this,
of the first derivative right over there.
And we've already seen examples of how can we
identify minimum and maximum points.
Obviously if we have the graph in front of us
it's not hard for a human brain to identify
this as a local maximum point.
The function might take on higher values later on.
And to identify this as a local minimum point.
The function might take on the lower values later on.
But we saw, even if we don't have the graph in front of us,
if we were able to take the derivative of the function
we might-- or even if we're not able to take
the derivative of the function-- we
might be able to identify these points as minimum or maximum.
The way that we did it, we said, OK,
what are the critical points for this function?
Well, critical points are where the function's derivative
is either undefined or 0.
This is the function's derivative.
It is 0 here and here.
So we would call those critical points.
And I don't see any points at which
the derivative is undefined just yet.
So we would call here and here critical points.
So these are candidate points at which our function might
take on a minimum or a maximum value.
And the way that we figured out whether it
was a minimum or a maximum value is
to look at the behavior of the derivative around that point.
And over here we saw the derivative is positive
as we approach that point.
And then it becomes negative.
It goes from being positive to negative
as we cross that point.
Which means that the function was increasing.
If the derivative is positive, that
means that the function was increasing as we approached
that point, and then decreasing as we leave that point, which
is a pretty good way to think about this
being a maximum point.
If we're increasing as we approach it
and decreasing as we leave it, then this
is definitely going to be a maximum point.
Similarly, right over here we see
that the derivative is negative as we approach the point, which
means that the function is decreasing.
And we see that the derivative is positive
as we exit that point.
We go from having a negative derivative
to a positive derivative, which means
the function goes from decreasing to increasing
right around that point, which is a pretty good indication,
or that is the indication, that this critical point is
a point at which the function takes on a minimum value.
What I want to do now is extend things
by using the idea of concavity.
And I know I'm mispronouncing it.
Maybe it's concavity.
But thinking about concavity, we could
start to look at the second derivative rather than kind
of seeing just this transition to think about whether this
is a minimum or a maximum point.
So let's think about what's happening
in this first region, this part of the curve
up here where it looks like a arc where it's opening
downward, where it looks kind of like an A
without the cross beam or an upside down U.
And then we'll think about what's
happening in this kind of upward opening U part of the curve.
So over this first interval, right
over here, if we start over here the slope
is very-- actually let me do it in a-- actually I'll do it
in that same color, because that's
the same color I used for the actual derivative.
The slope is very positive.
Then it becomes less positive.
Then it becomes even less positive.
It eventually gets to 0.
Then it keeps decreasing.
Now it becomes slightly negative, slightly negative,
then it becomes even more negative,
then it becomes even more negative.
And then it looks like it stops decreasing right around there.
So the slope stops decreasing right around there.
And you see that in the derivative.
The slope is decreasing, decreasing, decreasing,
decreasing until that point, and then it starts to increase.
So this entire section right over here,
the slope is decreasing.
And you see it right over here when we take the derivative.
The derivative right over here, over this entire interval
is decreasing.
And we also see that when we take the second derivative.
If the derivative is decreasing, that
means that the second derivative, the derivative
of the derivative, is negative.
And we see that that is indeed the case.
Over this entire interval, the second derivative
is indeed negative.
Now what happens as we start to transition
to this upward opening U part of the curve?
Well, here the derivative is reasonably negative.
It's reasonably negative right there.
But then it's still negative, but it
becomes less negative and less negative and less negative,
less negative and less negative, and less negative.
Then it becomes 0.
It becomes 0 right over here.
And then it becomes more and more and more positive.
And you see that right over here.
So over this entire interval, the slope or the derivative
is increasing.
So the slope is increasing.
And you see this over here.
Over here the slope is 0.
The slope of the derivative is 0.
The derivative itself isn't changing right at this moment.
And then you see that the slope is increasing.
And once again, we can visualize that
on the second derivative, the derivative of the derivative.
If the derivative is increasing, that
means the derivative of that must be positive.
And it is indeed the case that the derivative is positive.
And we have a word for this downward opening
U and this upward opening U. We call this concave downwards.
Let me make this clear.
Concave downwards.
And we call this concave upwards.
So let's review how we can identify
concave downward intervals and concave upwards intervals.
So if we're talking about concave downwards,
we see several things.
We see that the slope is decreasing.
Which is another way of saying that f
prime of x is decreasing.
Which is another way of saying that the second derivative must
be negative.
If the first derivative is decreasing,
the second derivative must be negative,
which is another way of saying that the second derivative
over that interval must be negative.
So if you have a negative second derivative,
then you are in a concave downward interval.
Similarly-- I have trouble saying
that word-- let's think about concave upwards,
where you have an upward opening U. Concave upwards.
In these intervals, the slope is increasing.
We have a negative slope, less negative, less negative, 0,
positive, more positive, more positive, even more positive.
So slope is increasing.
Which means that the derivative of the function is increasing.
And you see that right over here.
This derivative is increasing in value,
which means that the second derivative over an interval
where we are concave upwards must be greater than 0.
If the second derivative is greater than 0,
that means that the first derivative
is increasing, which means that the slope is increasing.
We are in a concave upward interval.
Now given all of these definitions that we've just
given for concave downwards and concave upwards,
can we come up with another way of identifying
whether a critical point is a minimum point
or a maximum point?
Well, if you have a maximum point,
if you have a critical point where the function is
concave downwards, then you're going to be at a maximum point.
Concave downwards, let's just be clear here,
means that it's opening down like this.
And when we're talking about a critical point,
if we're assuming it's concave downwards over here,
we're assuming differentiability over this interval.
And so the critical point is going
to be one where the slope is 0.
So it's going to be that point right over there.
So if you're concave downwards and you
have a point where f prime of, let's say, a is equal to 0,
then we have a maximum point at a.
And similarly, if we're concave upwards,
that means that our function looks something like this.
And if we found a point, obviously a critical point
could also be where the function is not defined,
but if we're assuming that our first derivative
and second derivative is defined here,
then the critical point is going to be one
where the first derivative is going to be 0.
So f prime of a is equal to 0.
And if f prime of a is equal to 0
and if we're concave upwards in the interval
around a, so if the second derivative is greater than 0,
then it's pretty clear, you see here,
that we are dealing with a minimum point at a.