WEBVTT
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we want to sketch the curve. Why is it
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into X times this word of two minus x squared
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. So in the structure they give us this laundry
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list of steps we should follow any time we want
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to sketch something, and so let's just go ahead
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and start with that. So the first thing we
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want to do is to determine our domain of our
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function. And so that's going to just be where
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, under our radical, it is strictly greater than
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or equal to zero. So not strictly bury them
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, but our eagle to greater than or equal to
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zero. So if we go ahead and factor the
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inside there, it might be easier to see what
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that should be. So is going to be the
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square root of two minus sechs times the square root
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of two plus X, and so from that we
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can conclude our domain. You should be the negative
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square root of two to the square root of two
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, including each of those points since we can,
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how under the radical equal to zero. The next
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thing we want to find is where our function is
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going to equal zero or our intercepts I should say
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so let's find the ex intercepts first, So X
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intercepts. So this is gonna be where zero is
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equal to X times two minus x squared. So
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looking at that factor form over there, that would
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be well X is equal to zero or X is
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equal to whatever makes underneath the radical equal to zero
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, which would be ex physical to plus or minus
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the square root of. And since X is equal
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to zero here, we know this is our why
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intercept as well at the origin. The next thing
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we want to look for is symmetry, so stuff
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like this isn't really known for being periodic, but
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we can at least check to see if it's going
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to be even or odd. So how the negative
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X is equal to negative X times. The square
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root of two minus will negative X squared is just
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going to be expired. So it's when the square
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and this here is equal to negative effort back.
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So we know our function is odd, and thus
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it has symmetry about the origin. So if the
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draw what's going on from 0 to 0 to square
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root of two we could just go ahead and reflect
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that across the origin to get the rest of the
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ground. What? Now that we have that we
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can go ahead and look att passin toots of this
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s troops. But there are going to be no
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acid toes because we're going to have no horizontal fascinated
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. Since we do not go to infinity and they
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and we have no vertical acidosis, there's nowhere in
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dysfunction that we're dividing by zero. So the next
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thing we want to look for so steps five and
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six is where the function increases and decreases, as
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well as any local maxes and men's we may get
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. So for that, we need to figure out
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what wide prime is going. So let's go ahead
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, get down another page. We're not squeezing things
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in, So why is he x times the square
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root to minus X work? So take the derivative
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This we're going to the team product will remember.
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Product rule says you take the 1st 1 will comply
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by the derivative of the second, and I'm going
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to rewrite that as a powder so two minus X
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squared to the one, huh? And then we
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need to add it in the opposite order. So
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to Maya's X squared times, the derivative of X
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. So take the derivative of two x minus minus
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expert to the one happen, going to need to
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use power and shingle. So it's going to be
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one house to minus X squared to the negative 1/2
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times the inside, which is going to be negative
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to X and the derivative of X rays. Just
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want to be one now for usable but algebra weekend
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. Rewrite this as two times one minus x weird
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over the square root oh, to minus X squared
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. Now we set this here equal to zero.
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That's going to tell us that one minus X.
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Where is Eagle? Deserve or X is equal to
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plus or minus one, So these are possible critical
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points and Millis. We also get critical points when
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our denominator is equal to zero or X is equal
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to plus reminds this word, but since these are
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endpoints, we don't actually need to be concerned with
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them. But if for some reason these weren't in
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points and the function was defined past this, then
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we would also need to look at those as well
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for critical values. All right, so let's go
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ahead and write that information down. So we have
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two times one minus x word over to minus X
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squared square root. Now we need to figure out
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where this function is increasing and increasing. So that's
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going to be where. Why Prime is strictly larger
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than zero, and this means the function is increasing
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. And I already went ahead and solved this beforehand
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. This being the interval negative 1 to 1,
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and to figure out where the function is decreasing,
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we want to see where wide prime is strictly less
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than and again already did this beforehand. And it's
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just going to be the rest of our domain that
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we have here. Excluding are in points so negative
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square root of two to negative one union, one
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to the square root. Now let's go ahead and
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look at that critical point that we had, which
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was are you actually have two critical points ex busy
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with the negative one and X is equal to one
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. So to the left of X equals negative one
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. The function is decreasing and after one, the
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function is decreasing and between negative one and one,
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The function is increasing. So what this tells us
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is X is equal to negative. One should be
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a local men and exiting Goto one should be eight
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local Max. All right, The last step before
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we actually start grabbing bits is to find our con
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cavity and any possible points of inflection. So we
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need to figure out what why double Prime is going
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to be. So let's go back over here now
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to take the derivative of this function we're going to
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need to use. Quotable. So why don't fine
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is going to equal to. So I'm gonna first
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factor that two out front so I don't have to
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carry it around and question Will says Low D hi
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and the numerator. We had one minus squared,
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I thought red there and then minus hi below.
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So them in the opposite order Thanks Oh, to
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minus X squared rooted all over our denominator squared.
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So that would be to minus X squared. And
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we would absolute value this but honor domain that will
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always be stricken, Arjuna zero. So we don't
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need to worry about that being zero. So now
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the derivative of one minus X squared what we said
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at the last part that that should be negative.
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Do X And we also found what the derivative of
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square root of to my sex squared is in the
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last part which would just be 1/2 to minus X
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squared to the negative 1/2 times negative to x and
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once again using a little bit of algebra. To
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simplify this down, we will get that this is
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too x expired minus three all over, two minus
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x squared to the three house help like And if
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we work, is that this equal to zero?
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We're going to get that X is equal to zero
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or X is equal to or zero is equal to
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X squared minus three, which would imply exit what
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plus or minus the square root of. But this
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here. So the square root of three is strictly
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larger than the square root of two. And negatives
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were 23 strictly less than the square root of too
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negative. So that tells us these are undefined points
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and we don't need to worry about them. So
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the only actual inflection point we get is exiting zero
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. And again we could set her denominator equal to
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zero. But we would just get that y double
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prime would be undefined at X is equal to plus
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or minus square root of two. And just like
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over here in the first part where we found our
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critical points for Why prime, since they're in points
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, we wouldn't really care about them being critical,
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since we can't really say what happened on the other
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side of the function. So only point of this
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in court. Inside of the second grave, it
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will be X is equal to zero, right?
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So, again we got our second derivative was two
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X over X squared minus three all over to minus
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X squared to the three house power and not a
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little bit. Now we need to figure out where
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the function is concave up, so they'll be calm
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. Cave up when y double private strictly larger than
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zero. And I would have been solved this already
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, and this should be on negative square root of
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2 to 0, at least over our domain.
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It should be that. And it's con cave down
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when y double prime district, less than zero or
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zero to the square root of now the only possible
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point infection that we had was X equals zero,
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and we can see to the left. The function
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is concave, but and to the right, the
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function is concave down. So since we have this
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change of kong cavity, we will know that that
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is a point of inflection. All right, so
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that was everything they told us we should do before
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we actually start sketching the curtain. So let's go
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ahead and start putting down our intercepts. So we
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haven't intercept at zero, and we haven't intercept at
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the square root of two and negative square root of
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two. We know the function is going to be
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symmetric about the origin, so we only really need
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to draw it from zero to square two. And
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then we can just sketch the curb in the opposite
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manner going, um, to the negative square.
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We have no accent. Oops. We know we
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have a max at exit one. So here is
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a max, and on the other side, negative
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one should be Amen. And we know what X
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is equal to zero. We should have a point
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of inflection. All right, so let's go ahead
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and start it ecstasy. Cool zero and go to
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the square root. So it is increasing until we
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hit one, because that's our maximum. So it's
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going to come over here and then fly and out
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at Exeter one. And then the next important point
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is it starts decreasing and we hit our intercept here
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and now we will just go ahead and reflect this
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across the origin, insuring that negative one. We
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have our minimum and be hit our intercept. So
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this year would be a sketch of our graph.
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And just to make sure so at excessive zero we
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do have are changing cavity because to the left is
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calm cable and to the right, it's Kong cave
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down. Now the only other thing you might want
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to possibly go do is to say, with this
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actual minimum value and maximum values should be. But
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other than that, since we're just sketching get,
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I think this is fine, since we show that
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it is a maximum at one as well as a
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minimum at exit