[MUSIC] Earlier, we saw how the sign, the 
S I G N, of the derivative, encoded 
whether the function was increasing or 
decreasing. 
Thinking back to the graph, here I've 
just drawn some random graph. 
What is the derivative encoding? Well, 
here at this point a, the slope of this 
tangent of negative, the derivative is 
negative, and yeah, the function's going 
down here. 
At this point B, the slope of this 
tangent line's positive, and the 
function's increasing through here. 
All right, the derivative's negative 
here, and it's positive here. 
The function's decreasing here and 
increasing here. 
So that's what the derivative is 
measuring. 
What is the sign of the second derivative 
really encoding? Maybe we don't have such 
a good word for it, so we'll just make up 
a new word. 
The sign, the sign of the second 
derivative, the sign of the derivative of 
the derivative, measures concavity. 
The word's concavity and here's the two 
possibilities. 
Concave up where the second derivative 
was positive and concave down where the 
second derivative is negative. 
And I've drawn sort of cartoony pictures 
of what the graphs look like in these two 
cases. 
Now note, it's not just increasing or 
decreasing but this concavity is 
recording sort of the shape Of the graph 
in some sense. 
Positive second derivative makes it look 
like this, negative second derivative 
makes the graph look like this and I'm 
just labeling these two things, concave 
up and concave down. 
And this makes sense, if we think of the 
second derivative as measuring the 
change, in the derivative. 
So let's think back to this graph again, 
here's this graph of some random 
function. 
Look at this part of the graph right 
here. 
That looks like the concave up shape from 
before where the second derivative was 
positive. 
So we might think the second derivative 
is positive here. 
That would mean that the derivative is 
increasing. 
What that really means is that the slope 
of a tangent line through this region is 
increasing, 
and that's exactly what's happening. 
The slope is negative here. 
And as I move this tangent line over, the 
slope of that tangent line is increasing. 
The second derivative is positive here. 
You can tell yourself the same story for 
concave down. 
So look over here in our sample graph. 
That part of the graph. 
Looks like this concave down picture, 
where the second derivative is negative. 
Now if the second derivative is negative, 
that means the derivative is decreasing. 
And yeah, the slope of the tangent line 
through this region is going down, 
right? The slope starts off pretty 
positive over here and as I move this 
tangent line over the slope is zero, and 
now getting more and more negative. 
So in this part of the graph, the 2nd 
derivative is negative. 
What happens inbetween? Where does the 
regime change take place? So over here, 
the second derivative is negative. 
Over here, the second derivative is 
positive. 
There's a point in between, maybe it's 
right here, and at that point, the second 
derivative is equal to zero and on one 
side, it's concave down and on the other 
side, it's concave up. 
A point where the concavity actually 
changes is called an inflection point. 
All right it's concave down over here and 
it's concave up over here and the place 
where the change is taking place, we're 
just going to call those points 
inflection points. 
It's not that the terminology itself is 
so important but we want words to 
describe the qualitative phenomenon that 
we're seeing in these graphs. 
Inflection points are something you can 
really feel. 
I mean, if you're driving in a car, 
you're braking. 
Right? That means the second derivative 
is negative. 
You're slowing down. 
And then, suddenly, you step on the gas. 
Now you're accelerating, and your second 
derivative is positive. 
What happened, right? Something big 
happened. 
[MUSIC] You're changing regimes from 
concave down to concave up. 
And you want to denote that change 
somehow. 
We're going to call that change an 
inflection point. 
[MUSIC]