WEBVTT
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All right, so we're gonna sketch occur for y
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equals X cubed minus 12 X squared plus 36 x
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. So in order to do that, we're going
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to start by finding its first derivative. So why
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Prime is equal to three x squared minus 24 acts
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. Pull us 36 and then we're going Teoh factor
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this and said it equal to zero to find the
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critical points so we can take out a three from
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all terms. So at least X squared minus eight
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X plus 12 and then that factors as X minus
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X minus to get a positive 12. And factors
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that multiply to 12 and add to eight are six
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and two. So we get the critical points are
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at X equal six and X equals two, and
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then we're going to find the second derivative. We're
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going to find the critical points of the second derivative
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, and then we are going to do a line
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test. So we're going to determine if our critical
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points are Max men's or nothing. So we need
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to look at a number below two on the line
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test so we could plug in zero. So zero
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minus something is negatives, you remind us something is
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negative. Negative times a negative is a positive terms
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of positive is a positive something between two and six
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. Like four positive negative positive. It's the positive
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negative. Positive would be a negative. Bigger than
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six would be 10. So positive. Positive.
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Positive is a positive. So that tells me the
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original is increasing to two, decreasing between two and
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six and increasing from six on so I could actually
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go down here and fill that in. So it's
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increasing from negative infinity to To and 60 Infinity.
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It's decreasing between two and six because of the negatives
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. First derivative. This is a polynomial, so
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it's domain is all rials. The X intercepts occur
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when Why is zero? So it's like the roots
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of the original LionOre sub secure when x zero.
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So in this case, zero. So that tells
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me on my graph I have the 00.0 What's nice
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is knowing that one of my roots is zero.
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I can use synthetic division to say one negative.
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12 0 36 10 Negative 12 00 36. So
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that means is we get X squared minus who factor
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. Okay, we used Listen a second. Is
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this even all it Or neither will again the That's
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why there is an X ray there. So you
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can actually factor out an X. And that would
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have left you with what I was trying to actually
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accomplish over there, which is that this is X
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squared minus 12 X plus 36. We know that
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that's a positive and a negative. So actually gonna
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have a repeated fruit here. This is gonna be
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X minus six X minus six, which means that
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one of the intercepts is at zero and the other
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is at six. And this is neither because it
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has a mixture of even and odd exponents. There
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are no vertical or horizontal Assam tunes. If we
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go back up here to the second derivative, we
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can use that dealt with Cohen Cavity and, um
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, points of inflection. So we're gonna look to
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numbers less than four. So zero So, like
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, is a negative numbers greater than four? So
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six of 36 minus 24 would be a positive.
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So what that means is on the original, it
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is concave down and concave up so down here filling
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in more data we have. Amen. When our
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graph goes from decreasing Teoh increasing so attacks equals six
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. We have a max when the graph goes on
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its first derivative from increasing to decreasing. So I
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X equals two. It is going cave up when
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the second derivative is positive. So four to infinity
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. It is Conchita down when the second derivative is
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negative. So negative infinity toe four. And we
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have a point of inflection at X equals four,
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which is where it changes from concave up to conchita
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down. So I know that from zero to to
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Mycroft is increasing and I could find out exactly where
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it's added to if I plugged to back in over
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here. So, um, plugging in to would
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give you two squared, which is four four time
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plus 36 is 40 and then minus 24 I would
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give you 16 times to give you 32. So
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, like craft would be like, way up here
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. So it increases to to decrease this from two
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to sex and then increases again and we can see
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that it changes calm cavity right there or and there
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is your craft off a curve. X cubed minus
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12 x squared, plus 36 acts