- [Instructor] Let F be a twice differentiable function.
One of these graphs is the graph of F.
One is of F prime.
And one is of the second derivative of F
F prime prime.
Match each function with its appropriate graph.
So I encourage you to pause the video
and try to figure out which of these is F,
which of these is F prime,
and which of these is F prime prime.
All right, now let's work through this together.
So there's a couple of ways to think about it.
One way to think about it is
if these were graphs of polynomials
the then F would be a higher degree polynomial.
So you have F, if you were to take the derivative of that
you'd get to F prime.
So this is derivative with respect to X.
And if you take the derivative with respect to X of that
you get F prime prime.
So, whatever degree polynomial this is
if it is a polynomial,
this will be a lower degree
and then this would be an even lower degree polynomial.
So let's see if we can make sense of these graphs
using that lens.
Well, one way to think about it is
higher degree polynomials are gonna have
more minima or more potential minimum
or maximum points.
And it looks like B here
has the most minima and maxima.
A has the second most.
And then C has the third most.
So my first hypothesis
is that
what we have right over here B
that this is the graph of F
that A is the graph of F prime.
And that C is the graph of the second derivative
F prime prime.
So that's just my hypothesis.
Now let's see if it actually makes sense.
And what I can do is
I can look at points where I know
that the derivative would be zero.
And then look at the derivative
to confirm what I think is the derivative
to confirm that they are.
And then I could also look at trends.
So let's say, if this is F, so let's say
and that was just my initial guess
that this is gonna be the function F.
So let's see, I have a slope of zero,
a slope of a tangent line of zero right over there.
So slope of zero right over there.
I have a slope of zero
right over there.
I have a slope of zero
right over there.
I have a slope of zero, a horizontal tangent line
right over there.
So let's see.
If this is the derivative
then this function should be equal to zero
at this X value,
this X value,
and this X value.
Because the slope of the tangent line
at those X values
looks like zero.
So this function is equal to zero here.
Here.
And, here.
Well these seem to coincide.
This coincides with this.
This coincides with that.
And that coincides with that.
And then that coincides with that.
So I'm already feeling pretty good about it.
So now let's see if
this F prime prime, what I think is F prime prime
is indeed the derivative of F prime.
So, it's got a minimum point right over here
it looks like the slope of the tangent line here is zero
let me do this in a different color.
So this looks like we have a slope
of zero there.
It looks like we have a slope of zero
right over there.
And it looks like we have a slope of zero
right
over there.
So at that point for that X value.
That X value.
And that X value.
And we see that this function is indeed
equal to zero, right over here.
Right over here.
And right over here.
At the exact same X values.
So, based on what I just said,
I would feel pretty good if I was under time pressure
I'd already feel pretty good about
how I'd match things up.
But you could also look at trends between points.
So for example
let me pick out a trend.
So, between this
between this maximum point
and this minimum point
at first,
the slope is decreasing.
So the slope is getting more
and more negative.
And you see from here
to here is indeed getting more and more negative.
And then it starts getting more and more positive
starting right about here.
So that trend seems to be consistent.
And so you could look for these trends, as well.
But I find looking at the minima and maxima
and where you see
horizontal tangent lines,
or where the slope of the tangent line is zero,
those are the easiest to test for your derivatives
or for your second derivatives.