WEBVTT

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What I have here in yellow is
the graph of y equals f of x.

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Then here in this
mauve color I've

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graphed y is equal to
the derivative of f

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is f prime of x.

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And then here in
blue, I've graphed

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y is equal to the second
derivative of our function.

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So this is the
derivative of this,

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of the first derivative
right over there.

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And we've already seen
examples of how can we

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identify minimum
and maximum points.

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Obviously if we have
the graph in front of us

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it's not hard for a
human brain to identify

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this as a local maximum point.

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The function might take
on higher values later on.

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And to identify this as
a local minimum point.

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The function might take on
the lower values later on.

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But we saw, even if we don't
have the graph in front of us,

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if we were able to take the
derivative of the function

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we might-- or even if
we're not able to take

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the derivative of
the function-- we

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might be able to identify these
points as minimum or maximum.

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The way that we did
it, we said, OK,

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what are the critical
points for this function?

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Well, critical points are
where the function's derivative

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is either undefined or 0.

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This is the
function's derivative.

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It is 0 here and here.

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So we would call
those critical points.

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And I don't see
any points at which

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the derivative is
undefined just yet.

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So we would call here
and here critical points.

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So these are candidate points
at which our function might

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take on a minimum
or a maximum value.

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And the way that we
figured out whether it

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was a minimum or
a maximum value is

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to look at the behavior of the
derivative around that point.

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And over here we saw the
derivative is positive

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as we approach that point.

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And then it becomes negative.

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It goes from being
positive to negative

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as we cross that point.

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Which means that the
function was increasing.

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If the derivative
is positive, that

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means that the function was
increasing as we approached

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that point, and then decreasing
as we leave that point, which

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is a pretty good way
to think about this

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being a maximum point.

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If we're increasing
as we approach it

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and decreasing as we
leave it, then this

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is definitely going
to be a maximum point.

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Similarly, right
over here we see

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that the derivative is negative
as we approach the point, which

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means that the
function is decreasing.

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And we see that the
derivative is positive

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as we exit that point.

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We go from having a
negative derivative

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to a positive
derivative, which means

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the function goes from
decreasing to increasing

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right around that point, which
is a pretty good indication,

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or that is the indication,
that this critical point is

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a point at which the function
takes on a minimum value.

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What I want to do
now is extend things

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by using the idea of concavity.

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And I know I'm
mispronouncing it.

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Maybe it's concavity.

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But thinking about
concavity, we could

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start to look at the second
derivative rather than kind

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of seeing just this transition
to think about whether this

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is a minimum or a maximum point.

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So let's think about
what's happening

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in this first region,
this part of the curve

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up here where it looks like
a arc where it's opening

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downward, where it
looks kind of like an A

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without the cross beam
or an upside down U.

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And then we'll
think about what's

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happening in this kind of upward
opening U part of the curve.

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So over this first
interval, right

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over here, if we start
over here the slope

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is very-- actually let me do
it in a-- actually I'll do it

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in that same color,
because that's

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the same color I used for
the actual derivative.

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The slope is very positive.

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Then it becomes less positive.

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Then it becomes
even less positive.

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It eventually gets to 0.

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Then it keeps decreasing.

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Now it becomes slightly
negative, slightly negative,

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then it becomes
even more negative,

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then it becomes
even more negative.

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And then it looks like it stops
decreasing right around there.

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So the slope stops decreasing
right around there.

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And you see that
in the derivative.

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The slope is decreasing,
decreasing, decreasing,

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decreasing until that point,
and then it starts to increase.

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So this entire section
right over here,

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the slope is decreasing.

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And you see it right over here
when we take the derivative.

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The derivative right over
here, over this entire interval

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is decreasing.

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And we also see that when we
take the second derivative.

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If the derivative
is decreasing, that

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means that the second
derivative, the derivative

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of the derivative, is negative.

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And we see that that
is indeed the case.

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Over this entire interval,
the second derivative

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is indeed negative.

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Now what happens as
we start to transition

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to this upward opening
U part of the curve?

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Well, here the derivative
is reasonably negative.

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It's reasonably
negative right there.

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But then it's still
negative, but it

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becomes less negative and less
negative and less negative,

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less negative and less
negative, and less negative.

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Then it becomes 0.

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It becomes 0 right over here.

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And then it becomes more
and more and more positive.

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And you see that
right over here.

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So over this entire interval,
the slope or the derivative

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is increasing.

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So the slope is increasing.

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And you see this over here.

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Over here the slope is 0.

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The slope of the
derivative is 0.

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The derivative itself isn't
changing right at this moment.

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And then you see that
the slope is increasing.

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And once again, we
can visualize that

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on the second derivative, the
derivative of the derivative.

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If the derivative
is increasing, that

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means the derivative of
that must be positive.

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And it is indeed the case that
the derivative is positive.

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And we have a word for
this downward opening

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U and this upward opening U.
We call this concave downwards.

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Let me make this clear.

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Concave downwards.

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And we call this
concave upwards.

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So let's review
how we can identify

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concave downward intervals
and concave upwards intervals.

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So if we're talking
about concave downwards,

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we see several things.

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We see that the
slope is decreasing.

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Which is another
way of saying that f

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prime of x is decreasing.

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Which is another way of saying
that the second derivative must

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be negative.

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If the first derivative
is decreasing,

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the second derivative
must be negative,

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which is another way of saying
that the second derivative

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over that interval
must be negative.

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So if you have a negative
second derivative,

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then you are in a concave
downward interval.

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Similarly-- I have
trouble saying

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that word-- let's think
about concave upwards,

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where you have an upward
opening U. Concave upwards.

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In these intervals, the
slope is increasing.

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We have a negative slope, less
negative, less negative, 0,

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positive, more positive, more
positive, even more positive.

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So slope is increasing.

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Which means that the derivative
of the function is increasing.

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And you see that
right over here.

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This derivative is
increasing in value,

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which means that the second
derivative over an interval

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where we are concave upwards
must be greater than 0.

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If the second derivative
is greater than 0,

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that means that the
first derivative

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is increasing, which means
that the slope is increasing.

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We are in a concave
upward interval.

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Now given all of these
definitions that we've just

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given for concave downwards
and concave upwards,

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can we come up with
another way of identifying

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whether a critical
point is a minimum point

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or a maximum point?

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Well, if you have
a maximum point,

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if you have a critical
point where the function is

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concave downwards, then you're
going to be at a maximum point.

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Concave downwards, let's
just be clear here,

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means that it's
opening down like this.

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And when we're talking
about a critical point,

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if we're assuming it's
concave downwards over here,

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we're assuming differentiability
over this interval.

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And so the critical
point is going

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to be one where the slope is 0.

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So it's going to be that
point right over there.

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So if you're concave
downwards and you

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have a point where f prime of,
let's say, a is equal to 0,

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then we have a
maximum point at a.

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And similarly, if
we're concave upwards,

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that means that our function
looks something like this.

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And if we found a point,
obviously a critical point

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could also be where the
function is not defined,

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but if we're assuming
that our first derivative

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and second derivative
is defined here,

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then the critical point
is going to be one

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where the first derivative
is going to be 0.

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So f prime of a is equal to 0.

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And if f prime of
a is equal to 0

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and if we're concave
upwards in the interval

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around a, so if the second
derivative is greater than 0,

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then it's pretty
clear, you see here,

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that we are dealing with
a minimum point at a.