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We're asked, at which points
on the graph is f of x times f prime of x equal to 0? So if I have the product of
two things and it's equal to 0, that tells us that at least
one of these two things need to be equal to 0. So first of all, let's see are
there any points when f of x is equal to 0? So we're plotting f of
x on the vertical axis. We could call this
graph right over here, we could say this is
y is equal to f of x. So at any point, does the y
value of this curve equal 0? So it's positive, positive,
positive, positive, positive, positive. But it is decreasing
right over here. Well, it's decreasing here. Then it's increasing. Then it's decreasing. And it does get to
0 right over here, but that's not one of
the labeled points. And they want us to pick
one of the labeled points or maybe even more than one
of these labeled points. So we're going to focus on where
f prime of x is equal to 0. And we just have
to remind ourselves what f prime of x
even represents. f prime of x represents the
slope of the tangent line at that value of x. So for example, f
prime of 0-- which is the x value for this
point right over here-- is going to be some
negative value. It's the slope of
the tangent line. Similarly, f prime
of x, when x is equal to 4-- that's what's
going on right over here-- that's going to be the
slope of the tangent line. That's going to be
a positive value. So if you look at
all of these, where is the slope of
the tangent line 0? And what does a 0
slope look like? Well, it looks like
a horizontal line. So where is the slope of the
tangent line here horizontal? Well, the only one
that jumps out at me is point B right over here. It looks like the slope
of the tangent line would indeed be horizontal
right over here. Or another way you could think
of it is the instantaneous rate of change of the function,
right at x equals 2, looks like it's pretty
close to-- if this is x equals 2-- looks like
it's pretty close to 0. So out of all of
the choices here, I would say only B looks like
the derivative at x equals 2. Or the slope of the tangent
line at B, it looks like it's 0. So I'll say B right over here. And then they had this kind of
crazy, wacky expression here. f of x minus 6 over x. What is that greatest in value? And we have to interpret this. We have to think about what does
f of x minus 6 over x actually mean? Whenever I see
expressions like this, especially if I'm taking a
differential calculus class, I would say well,
this looks kind of like finding the slope
of a secant line. In fact, all of what we
know about derivatives is finding the limiting value
of the slope of a secant line. And this looks
kind of like that, especially if at some point,
my y value is a 6 here. And this could be the
change in y value. And if the corresponding
x value is 0, then this would be f of
x minus 6 over x minus 0. So do I have 0, 6
on this curve here? Well, sure. When x is equal to 0, we see
that f of x is equal to 6. So what this is right over
here-- let me rewrite this. This we could rewrite as f
of x minus 6 over x minus 0. So what is this? What does this represent? Well, this is equal
to the slope-- let me do some of
that color-- this is equal to the slope of the
secant line between the points, x, f of x, x, and whatever
the corresponding f of x is. And we could write it as 0,
f of 0 because we see f of 0 is equal to 6. This right over here is f of 0. In fact, let me just
write that as 6. And the point, 0, 6. So let's go through
each of these points and think about what the
slope of the secant line between those points are and
point A. This is essentially the slope of the secant
line between some point x, f of x, and essentially point
A. So let's draw this out. So between A and B you have
a fairly negative slope. Remember we want to
find the largest slope. So here it's fairly negative. Between A and C,
it's less negative. Between A and D, it's
even less negative. It's still negative,
but it's less negative. And then between A and E, it
becomes more negative now. And then between A and F, it
becomes even more negative. So when is the slope of
the secant line between one of these points
and A the greatest? Or I guess we could
say the least negative? Because it seems like
they're always negative. It would be between
point D and A. So when is this
greatest in value? Well, when we're
looking at point D. At point D, x is
equal to 6 and it looks like f of x is like
5 and 1/2 or something. So this will turn into f of
6, which is 5 and 1/2 or maybe it's even less than
that-- 5 and 1/3 or something, minus
6 over 6 minus 0. That's how we'll
maximize this value. This is the least negative
slope of the secant line.