WEBVTT
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So for the domain, we know that it's gonna
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be, um X is greater than zero because it
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has to be first of all, ex cannot be
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zero because of first part. And the second part
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tells us that it has to be positive. So
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x is greater than zero intercepts. There are none
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symmetry, there is none. And for Assam totes
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, if we look for a horizontal, ask himto
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, for example, we have to set limits.
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Um, as X approaches infinity or limit as X
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approaches. Negative infinity. We see that there's no
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horizontal. However, there is a vertical ass until
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at X equals zero. Because we set our denominator
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for one of her ex. We said the denominator
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to zero. So x zero is our, ah
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, vertical ass in two. And now we look
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for increasing decreasing intervals. So you find the first
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derivative y prime, and that is one of her
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ex minus one over X squared. And if we
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set our first derivative to zero, we confined critical
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points and we see that X is one is a
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critical point. So if we do a first derivative
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test, so this is for F prime and we
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substitute, um, and research for inside of our
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derivative. We're gonna plug values that are smaller than
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one. What we have to remember to stay in
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our domain. So if it's smaller than one,
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it's going to be negative. If it's greater than
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one, it's gonna be positive. So that means
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that it's decreasing until it gets to one and increasing
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after that. So if it's going from negative to
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positive, so decreasing to increasing its gonna be men
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a local minimum value. And if you want to
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know the why value of our critical point we go
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F one. So we're putting it back into her
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original equation to find one. So 11 is our
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minimum local minimum value for our graph. And now
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we can find, um, we can look at
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kong cavity for this graph by finding the second derivative
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. So why double prime equals negative, and this
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negative is on the outside of the entire fraction.
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So negative X minus two over X cubed. So
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if we set that equal to zero, we see
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that X is equal to two and now our second
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derivative test, so double prime, and this is
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too and we have to make sure to test values
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in our domain. So, for example, if
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it says 0.5 for over here so smaller than two
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. But in still inside of her domain, we're
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going to get positive. So Khan came up,
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and beyond that, it's gonna be Khan came down
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so negative. Okay? And now we can graph
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with the information that we got. So first thing
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I'm gonna put on the graph is the, um
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, vertical ascent. Tow X equals zero, and
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we can see that, um, we have a
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minimum value at 11 So let's say this is one
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. This is one. So that's our local minimum
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value. And, um, we see that we
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haven't Oh, yeah. This is an inflection point
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because it's switching signs from Khan gave up to conquer
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it down. So we can say that this is
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an inflection point at two. All right, so
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, um, say this is about to So this
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is a rough sketch of our graph. It's gonna
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be decreasing until it hits our minimum value so it
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cannot touch the ass in tow. So decreasing until
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hits the minimum value, then it's gonna increase from
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then on. So it's so we have con cave
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up. That's correct. And then it's gonna be
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calling came down after it gets to our inflection permit
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. So it's Kong cave up, but still increasing
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until it's our inflection point that it's gonna be con
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cave down but still increasing, and that's a rough
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sketch.