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Use the guidelines of this section to sketch the curve.$$y=x^{3}-12 x^{2}+36 x$$

$(4,16)$

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 4

Curve Sketching

Derivatives

Differentiation

Applications of the Derivative

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

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