- [Instructor] The twice
differentiable function G
and its second derivative
G prime prime are graphed.
And you can see it right over here.
I'm actually working off of the article
on Khan Academy called
Justifying Using Second Derivatives.
So we see our function G.
And we see not its first derivative
but its second derivative here
in this brown color.
So then, the article goes on to say,
or the problem goes on to say,
four students were asked
to give an appropriate
calculus-based justification for the fact
that G has an inflection point
at X equals negative two.
So let's just feel good
that at least intuitively
it feels right.
So X equals negative two, remember what
an inflection point is.
It's where we're going
from concave downwards
to concave upwards.
Or, concave upwards to concave downwards.
Or another way to think about it
it's a situation where
our slope goes from
decreasing to increasing,
or from increasing to decreasing.
And when we look at it
over here, it looks like
our slope is decreasing, it's positive,
but it's decreasing, it goes to zero.
Then it keeps decreasing, it becomes
it's negative now.
It keeps decreasing until we get to about
X equals negative two
and then it seems that it's increasing,
it's getting less and less
and less and less negative.
It looks like it's a zero right over here
then it just keeps
increasing, it gets more
and more and more positive.
So it does, indeed, look
like at X equals negative two
we go from being concave downwards
to concave upwards.
Now a calculus based justification
is we could look at its,
at the second derivative
and see where the second derivative
crosses the X-axis.
Because where the second
derivative is negative
that means our slope is decreasing
we are concave downwards.
And where the second
derivative is positive
it means our first derivative
is increasing, our slope
of our original function
is increasing and we are concave upwards.
So notice, we do indeed,
the second derivative
does indeed cross the X-axis
at X equals negative two.
It's not enough for it to just be zero
or touch the X-axis,
it needs to cross the
X-axis in order for us
to have an inflection point there.
So given that, let's look at the students
in justifications and see
what we can, if we can kind of play
put the teacher hat in our mind
and say what a teacher would say
for the different justifications.
So the first one says,
the second derivative of G
changes signs at X equals negative two.
Well that's exactly what
we were just talking about.
If the second derivative changes signs
in this case, it goes
from negative to positive,
that means our first derivative went from
decreasing to increasing.
Which is indeed, good for saying this is a
calculus-based justification.
So, at least for now I'm gonna put
kudos you are correct there.
It crosses the X-axis.
So this is ambiguous.
What is crossing the X-axis?
If a student wrote this I'd say,
what are they talking about, the function,
are they talking about
the first derivative,
the second derivative.
And so I would say, please
use more precise language.
This cannot be accepted as
a correct justification.
I will read the other ones.
The second derivative of G is
increasing at X equals
negative two.
Well no, that doesn't justify
why you have an inflection point there.
For example,
the second derivative is increasing
at X equals negative 2.5.
The second derivative is even increasing
at X equals
negative one.
But you don't have an inflection point
at those places.
So I would say, this doesn't justify
why G has an inflection point.
And then the last student response,
the graph of G changes concavity
at X equals negative two.
That is true
but that isn't a
calculus-based justification.
We'd want to use our
second derivative here.