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2x+6 point) Consider the function f(x) 61+2 For this function there are two important intervals: (~0, A) and (A, * where the function is not defined at AjFindFor ea...

Question

2x+6 point) Consider the function f(x) 61+2 For this function there are two important intervals: (~0, A) and (A, * where the function is not defined at AjFindFor each of the following intervals, tell whether f(x) is increasing (type in INC) or decreasing (type in DEC): (~m,A):(A,0 Note that this function has no inflection points but we can still consider its concavity: For each of the following intervals, tell whether f(x) is concave up (type in CU) or concave down (type in CD): (~m,A):(A, *

2x+6 point) Consider the function f(x) 61+2 For this function there are two important intervals: (~0, A) and (A, * where the function is not defined at Aj Find For each of the following intervals, tell whether f(x) is increasing (type in INC) or decreasing (type in DEC): (~m,A): (A,0 Note that this function has no inflection points but we can still consider its concavity: For each of the following intervals, tell whether f(x) is concave up (type in CU) or concave down (type in CD): (~m,A): (A, *



Answers

Find: (a) the intervals on which f is increasing, (b) the intervals on which f is decreasing, (c) the open intervals on which f is concave up, (d) the open intervals on which f is concave down, and (e) the x-coordinates of all inflection points. $f(x)=(2 x+1)^{3}$

So this function is increasing. Um, from 2.2963 Um And then it's decreasing on the intervals from negative infinity to zero and 00.29632 Positive infinity. The times It's Kong Cave. Ah, con cave up would be never because, as we see here, there is this kink right here. It's not differential at this point. Um, but everywhere, uh, the derivative function, as we see is, has a negative slopes. That means that it's on Li con cave down from negative infinity to infinity. So there's no conclave up, and there's no inflection point.

Functional analysis on this. We get that 1st. We need to find her preservative Secrets for three times x plus two squared. So this is equal when we set it equal to zero here. Get that X is equal to -2. Doing a search shirt with them get they're both positive because anything second power is positive there. So therefore it's increasing from negative infinity to to union too to infinity. Okay. So it's never decreasing. Let's look at the next part. So let's take the second derivative here. That's equal to six times x plus two. So that we set that equal to zero here. We get also X equals negative to hear again. So supposed to be negative too. So increasing for negative to -2. Union -2 to Infinity. Just bring out the negative there. Alright, So never decreasing. And then here, so if we look at X equals negative two. All right. Still sign chart for that on the second derivative, We see that to the left of -2, it's going to be negative, and to the right of it, it's going to be positive. And so therefore you have an inflection point At x equals -2. You also can tell that it's going to be so concave town from negative infinity to negative two and came up trump -2 to Infinity. It's for the Science fair that we had up here. Right for second truth. That's it.

So we're doing a fresh analysis. Let's first simple planets. We have excellent. Okay. Rex to the 1/3. Detective Dorado. This. We have paramedics single too. For third sex. To the one third plus four thirds Next to the -2 or three. So setting that equal to zero we get. So since the four thirds are the same here, you can ignore those. We have X to the one third. It's equal to you think it gave extended two or three. So that simplifies to next to the 1/3 equals one over two thirds. Cross multiplication. We get X. It's equal to 21 so X is equal to negative. So X is equal to negative one here. Okay, so from here let's do a science right here test negative one. So I see that to the write it up, it's going to be all positive to let the that's all negative. All right. That means that it's going to be increasing from -1 to Infinity. You decreasing from Take it if it's a -1. Okay, take a look at the second derivative shingles four lengths Extent -2/3 minus he notes Next to Native five Threats. Okay, sorry I said that equal to zero. So you have 496 to 9 to two thirds. All right. So because they're both over nine here, I can ignore the ninth there. The dominator and I have Eat X to the negative 5/3 Cause four X donated 2/3. Do I have a four to both sides here? I got to be to rex to the thirds. Course one over two thirds. Which means that when we simplify this out here, we get X to the five thirds minus two X to the two thirds. Zero Factor out of 2/3 here, I'm not a thanks -2. Equalling zero. That means I have X equals zero and the next equals two. Still side chart for that. So we see that if we plug in say like one, I got a negative in there than everything else. Three or negative one. Yeah, positive positive. So that tells us it's going to be Kharkiv up trump negative affinity 80 to to infinity is going to be concave down from 0 to 2 and that we have an inflection point At x equals zero and At x equals two. That's it.

So looking at a graph here, we can look at the, um, derivative crafted, get a better view of things. So we see that, um, here that the graph is increasing from 0.25 to infinity and and it's decreasing from negative. Infinity 2.25 Then we see that, uh, it's Kong cave up. The graph is Kong. Keep up. So here it becomes the graph goes the negative. We're assuming here and we see that the graph is conch it up. Um, all the way from here. So from 02 infinity, the graph is Khan gave up and from negative infinity to zero. Ah. So the graph is Khan gave up from negative infinity to negative 0.5 and then from negative 0.52 zero, it's concave down and then from zero to infinity, its conclave up. So the inflection point to be a negative 0.0.5 and zero


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