WEBVTT
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we want to sketch a graph of why is he
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with that excuse? Plus three x squared. So
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in the shocker against us, a laundry list of
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how we should go about doing this. So Step
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one was first to identify our domain. And we
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know polynomial is have a domain of all real numbers
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, just negative entity to infinity. Then it suggests
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we move on to our intercepts. So let's go
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ahead and start with the ex intercepts. So ex
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intercepts. Well, this means why is equal to
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zero. So zero is equal to x cubed plus
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three x squared and noticed We compact out of X
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squared. Thanks, Bus three. And this tells
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us X is equal to zero or ex busy,
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too. Negative. Three. So we have our
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duplex values or X intercept. No, let's go
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ahead and find out why. Intercept. Remember,
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this is we set X equal to zero, so
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you might be able to notice that from movie did
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in the last part, we should be able to
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tell, but we could just go out and plug
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this end. So zero cubes zero and then zero
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squared to zero. So we get our wide value
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of zero for the wide, interesting. Next.
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It wants us to check for any kind of symmetry
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. So we want to figure out the functions,
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even or odd or anything like that. So let's
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go ahead and put a little biter here. So
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in the tree. So if this year is supposed
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to be EPA Becks, let's go ahead and look
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at what of negative exes. There's gonna be negative
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x cute. What's three negative X squared, which
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is going to be negative, X cubed, plus
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pretty X squared. So since this does not equal
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to EPA, Becks or negative at the vex,
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we have no symmetry. Next, it wants us
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to find acid trips. So if we take the
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limit so the minute as X goes to infinity of
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f of X. Well, since this is a
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polynomial, we know that this is going to go
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to infinity and the limit as X approaches. Negative
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infinity. Oh, that Lex. Since it's an
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odd polynomial, we know this is going to go
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to negative infinity. So no asking those least we
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now know in next we should find where the function
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is increasing such decreasing, so increasing slash decreasing So
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this says we need to look at the second derivative
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. They're the first word. So we get wide
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prime is going to be derivative with respect to X
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Oh, X Q. What excuse plus three X
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squared. Now we can go ahead and used parable
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to take the derivative of each of those. So
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be it. Three Next squared bluff six six x
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not sex 66 six x and then to the first
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power. Now, when this is strictly less than
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zero, it'll be Do you think I want it
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stripped? The grand jury will be increasing. So
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let's go ahead after this role so we can pull
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out three end of X, We get X plus
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one like this that you're here to ex exposed to
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and we know I just kind of looking at this
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. This is a quadratic. So it should look
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something like this. Insolent coefficients. Positive. Just
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using that. We can say that wide prime will
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be strictly greater than zero wet. We are on
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the interval. Negative infinity two negative too. Union
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zero to infinity. And then why prime will be
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strictly less than zero are decreasing on negative 2 to
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0 the next thing they want us to find is
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our local max and open beds. So to do
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that, we want to set that equal zero so
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local max slash minimum. So we want to look
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at when why Prime is equal to zero and looking
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at what we have here, we can see that
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it will be possibly at X is equal to zero
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, and X is equal to negative too. And
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we can use this information here that we found about
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the increasing and decreasing intervals to see that we are
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increasing into. I hope so. My fellow out
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there so negative too so on the left of negative
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two were increasing into it and then decreasing. Actor
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. So this is going to be a local Max
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and X is equal to zero. What? Decreasing
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to the left of it and increasing to the right
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. So this would be a men and this will
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be a backs. So we have that the next
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thing will want to do is to find Kong cavity
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and points of infection. So I'll go ahead and
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have a little bit so khan cavity slash inflection once
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. So this means we need to look at the
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second derivative. So why double crime? Well,
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over here we found that wide Prime iss three ex
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swearing off sex sex. So taking the derivative of
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this function. So deed by the x of three
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X squared plus six x So they used power once
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again. So we would get six x to the
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first power plus and in derivative of X, it's
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just one. So we get six there and but
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we want to find changing con cavity. But we
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know that this here is just going to be a
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line that looks like that. So if we said
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this equal to 0666 um, we can just divide
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everything by sexual past and then subtract one over,
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we'll get negative one. Is it? So at
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this point here, who have a change in con
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happy And we also know just using this function here
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. So when why? Double prime is strictly greater
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than zero. Um, this will be calm,
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Cape up call on, and this occurs using the
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fact that we know the zero of this line is
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exceeded negative one, and it's increasing. That's it
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won't be from negative one to infinity And then why
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double promise like to be a messenger or con down
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on the other intervals. So negative infinity too negative
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. All right, So now that we have all
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this, it's just not one more time and start
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graphing everything. So you go ahead, Mark a
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little area right here. 1st 2 breaths. So
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go ahead and first plot all of our relevant information
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. So we know our ex intercepts were going to
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be at zero and negative debris, So zero and
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123 negative three. And we also know that s
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R Y intercept is also at 00 So we don't
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have to do anything with those, um, then
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going to symmetry. While there was no symmetry asking
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Taub's there were no Assam toes, but we know
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are in behavior is going to be to infinity on
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the right and to negative infinity on the bus,
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uh, intervals of increasing and decreasing. We have
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that, but more importantly, we know we have
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a minimum at zero at a maximum, innit?
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Negative too. So this is going to pass through
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like this, come up until we hit some value
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for negative to come back down, and then it's
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going to touch this point because we have no other
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ex intercepts. So you have to bounce off that
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point. They come back up like that. So
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that uses five and six. Because we have our
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intervals of increasing and decreasing calm cavity at X equals
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negative one, which would be this point here.
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We can see that's going from con cave down on
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the left to con cave up on the right.
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And now the only thing we might want to do
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is to possibly find out what are local Maximum is
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right here. So to do that, we're just
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going to look at what, uh, of negative
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, too. Is so that's going to be negative
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eight plus well or four. So that point there
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would be negative to four. So this would be
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a nice little sketch of the graph using the method
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outlined in this chapter.