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(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the inforvation from parts (a)-(c) to sketch the graph. Check your work with a graphing device if you have one.$f(x)=36 x+3 x^{2}-2 x^{3}$

A. decreasing: $(-\infty,-2),(3, \infty) \quad$ increasing: $(-2,3)$B. $\operatorname{Min}(-2,-44) \operatorname{Max}(3,81)$C. Concave up: $\left(-\infty, \frac{1}{2}\right)$Concave down: $\left(\frac{1}{2}, \infty\right)$Inflection points: $x=\frac{1}{2}$D. SEE GRAPH

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 3

Derivatives and the Shapes of Graphs

Derivatives

Differentiation

Applications of the Derivative

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for this question. We have four different parts. Before doing that, we just write it out the narrative and the second, the third. Because we're gonna use this to functions. And if you if I write doubts this first of the directive, we can just use the, um you could just factor this into some senior points. So, like this, it's easier to figure out their roots off this function if we affect her in this way. And the philosophical maturity, if we have just minus 12 explosive six. So it's already in here. Okay, so for Patty, we are looking for the interval off. Increase of decrease on. So first, just let a prime of X equals zero. So we have tools. Why is minus two the otherwise three since the Romanoff Fe's ah, from minus infinity to positive? If anything, we have the race up intervals the first wise from minus your affinity to minus two. So only seen terrible if prime affects is, um, Conectiv uniformly negative. So whenever, um, no matter which number you choose the inside this interval f prime off X, the first solidarity of is always negative. That means only sent her boys decreasing and with the same thing for the rest to interval. So next wise, from next year, 2 to 3. So over over this integral the first that the first of the purity of its positive safety increasing and from three to positive infinity on the first of the curative is negative again. So it's decreasing now We are ready to through the poppy, the properties related to party. So once we have the product a, um, we know that there is a local oh, maximum at X equals with three because, um, on the left off, three functions in crazy and on the right is decreasing. So there's a peek at X equals three riches and local max and then the value is just plugging three. Into that function, we have 81 and the for local minimum. We have a local minima at X equals two minus two rental value. Yes, uh, minus 44. So for Passy, we're gonna use the second of the narrative. So first we set the second of the curative. He cost zero. So we have X equals to 1/2. This is only one roots. And the by using this points, we can splice the domain to two intervals. So the first ones for a negative into you too 1/2. The 2nd 1 is from what have toe qualitatively infinity. So over the first interval, the saco the purity of four ff we can check it actually called him that music. It's concave up for it. And for the next interval, the Saco maturity If it's money for new negative, So it's country downwards and this is the community, um, change changes that, um, X equals one have. So we have onion, so we have And the inflection points at X equals 1/2 because I can cavities are different. Um, for two sides off, um, X equals 1/2. So for party, we're gonna use all these informations to sketch the graph The effect The graph looks like this. So we have a coordinate. So we label some point here, so it's minus two here in the politic to here, and we have one have Yeah. So the graph of the function looks like this, right? So he looks like, um So we need to be careful. Publican carry t here. No, you put something that lets so we can see that we have a critical point and it's a local Milliman. We have a quick a point here. You tell a local maximum. Also, we have any inflection points at X equals don't have and be careful effect for the function. We actually know the Y intercept, which is zero. So with my meat years off this information to get this graph and it's a sketch, it's known The acreage, for example, is mark. When X equals two minus two, if X should be minus 44 when X equals the to if affects should be thank you one and leads it.

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