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Intro music.
Gabriel: Hi I'm Gabriel. I'm here with Ben and he is going to help me with this problem.
What are we working on today?
Ben: Today we are going to find where a function is increasing and decreasing as well as where
it's concave up, concave down, and the points of inflection.
Gabriel: Okay. Perfect. So that means that we need to take the first derivative of this
function. Which would be f prime of x is equal to 8 x to the third
minus 64 and we need to find the critical numbers. to do that we need to set this equal
to zero. Is that right, Ben?
Ben: that's right.
Gabriel: now I like to stop here. especially when you have something really funky here
and if you notice this is a three. right? and what can we do
when we have a three here? what type of equation is that?
Ben: it looks like we have a difference of cubes. So we have to use our formula. Well
actually if it is a difference of cubes and all we are trying to
find is the critical numbers we can start by moving the number over. the 64. And start
solving for x.
Gabriel: Perfect. So we would first add 64 plus 64 those would cancel and we would be
left with 8 x cubed is equal to 64. and then we would divide by 8?
Ben: We divide by eight. Yes.
Gabriel: Those eights cancel and now we are stuck in here. 64 goes into 8?
Ben: 8 times.
Gabriel: 8 times. and then what would we do here?
Ben: In order to get rid of the cubed we have to take the cubed root.
Gabriel: the cubed root. yes. That three would cancel that cubed root and we would get x
is equal to 2. and that is, ladies and gentlemen, our critical
number. What do we do with that?
Ben: For the critical number. we need to plot it on a number line and find test points on
either sides of it.
Gabriel: Cool. And what color should our number line be?
Ben: We can do it in yellow.
Gabriel: Alright. So say the number line starts from here and then goes this way so that infinity
is over here and negative infinity is over here.
In this case we only have one critical number in our first derivative. so that is possible.
we are just going to have one critical number. so that
means we are going to have two test points. And zero and three would be our numbers. What
do you think?
Ben: Sounds good to me.
Gabriel: Now. To check whether it's increasing from negative infinity to 2 we need to get
our test points and plug them into ... where?
Ben: Our first derivative.
Gabriel: And what color is that going to be?
Ben: We are dealing with ... well i guess we can keep it in orange. Since it's one of
our test points.
Gabriel: So this is going to... We are going to have f prime of some number here which
is going to be orange 8 times our test point minus sixty four and we are going to do this
twice. Right? once for our first test point then other for our second test point. And
our test points are in orange. So this is zero. This is going to be zero. Zero. I'm
sorry I lie this is going to be three and this going to be three. And so what do you
get? Use your handy dandy calculator and let us know the evaluation of what this is.
Ben: Okay. For the first point. When we plug in zero the first term is going to cancel
because any thing times zero is just zero. So we are going to get negative sixty four
for our first test point.
Gabriel: so this is negative sixty four. And so this is going to be three cubed. Let see
if we can beat Ben. Come on. Three cubed. What is that? Ah he beat us? Dang it.
Ben: I get one fifty two.
Gabriel: one fifty two. So its decreasing from negative infinity to two and from two
to infinity is increasing.
Ben: And why is it decreasing and increasing?
Gabriel: Well Ben good question. The reason why its is decreasing is because a negative
slope ... In this case the first derivative can be thought of as a slope. Since we have
a negative slope here then anywhere between negative infinity and two we are going to
be decreasing. And that's the reason why. So the same thing follows for this. Since
this is a positive slope then from the region from 2 to infinity we would be increasing
in that roller coaster. so this would be the valley of the roller coaster. I love roller
coasters.
Ben: ha ha!
Gabriel: All my analogies revolve around roller coasters and
Ben: Cats.
Gabriel: Puppies. I like puppies.
Ben: Okay.
Gabriel: Now. So we are done with the first derivative. Almost. Except for one thing.
We need it to take the second derivative.
Ben: right
Gabriel: So lets go back and see what our first derivative was.
Ben: This is moving on two part b right?
Gabriel: Yes. Let's use red.
Ben: Sure. Gabriel: The second derivative is going to
be. What is eight times three?
Ben: twenty four.
Gabriel: Yes. Thank you. 24 x squared and 64 that's a constant so that goes away. So
we are just left with that. Now just like we did before we need to find our critical
numbers but this time this critical numbers are going to be for our second derivative.
And that is going to be minus 24 x square equal to zero.
Ben: minus?
Gabriel: I lie. That should be a positive.
Ben: nice safe.
Gabriel: Nice safe. Thanks Ben. So what do we do here.
Ben: looks like, in order to solve for x, we need to divide by 24 from both sides.
Gabriel: Yes. First we divide both sides by 24. Now what is x square.
Ben: You mean zero.
Gabriel: Yeah this is zero. Dang. Ben. That's why I have Ben here he keeps me straight.
Let's say this is zero. Right? And what do we have over here. Negative infinity? Because
that is to our left hand side and this is going to be positive infinity. And lets use
orange again. Those are going to be our test points. What should we test? Negative one
and one?
Ben: sounds good. keep them small.
Gabriel: And we are going to plug those test into our first derivative? No.
Ben: Second Derivative.
Gabriel: We are not we are going to plug them into our second derivative.
Ben: Ah. You tricky.
Gabriel: Yeah. This is going to be negative one and we are going to plug that into our
second derivative. What does that give. What is twenty four times negative one. So we are
going to first square negative one. That is going to become a positive one and that is
going to be a positive 24 for both cases. So we are going to be concaving from this
region. From negative infinity to zero. If I change this to a plus one well that is not
going to change much because if I square one I get one. One times 24 is 24 again. So I'm
also going to have a positive concavity between zero and infinity. So in this case we wouldn't
have an inflection point. Because, remember, when do we have an inflection point again?
Ben: When we are changing from. We are changing concavity from up to down and down to up.
Gabriel: In this case that wouldn't happen. So now how would we label the intervals. Lets
label them in purple. So I would say f of x is decreasing from what range to what range?
Ben: Looking at first derivative number line it looks like it is decreasing from negative
infinity to positive two.
Gabriel: Negative infinity to positive two. And f of x is increasing from?
Ben: just on the other side. From positive two to positive infinity.
Gabriel: What about this right here. How would we describe the concavity. So we could say
concave up from where to where?
Ben: I would probably say from negative infinity to infinity.
Gabriel: negative infinity to infinity. And that is it!