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7. [-/5 Points]DETAILSLARCALCII 3.3.019.My NotESASK YOUR TEACHERPRACTICE ANOTHERConsider the following function: f(x) (a) Find the critical numbers of f. (Enter you...

Question

7. [-/5 Points]DETAILSLARCALCII 3.3.019.My NotESASK YOUR TEACHERPRACTICE ANOTHERConsider the following function: f(x) (a) Find the critical numbers of f. (Enter your answers as a comma- separated list. )(b) Find the open intervals on which the function is increasing or decreasing: (Enter your answers using interval notation. If an answer does not exist enter DNE:) increasingdecreasing(c) Apply the First Derivative Test to identify the relative extremum: (If an answer does not exist, enter DNE: )

7. [-/5 Points] DETAILS LARCALCII 3.3.019. My NotES ASK YOUR TEACHER PRACTICE ANOTHER Consider the following function: f(x) (a) Find the critical numbers of f. (Enter your answers as a comma- separated list. ) (b) Find the open intervals on which the function is increasing or decreasing: (Enter your answers using interval notation. If an answer does not exist enter DNE:) increasing decreasing (c) Apply the First Derivative Test to identify the relative extremum: (If an answer does not exist, enter DNE: ) reiative maximum Tclutivc minimum Nood Holp? Raud I Maleht



Answers

Answer the following questions about the functions whose derivatives are given. a. What are the critical points of $f ?$ b. On what open intervals is $f$ increasing or decreasing? c. At what points, if any, does $f$ assume local maximum or minimum values? $$f^{\prime}(x)=(x-7)(x+1)(x+5)$$

Okay, so let's start shaking. Are critical points or finding are critical points. So we re taking preservative. So we have four times six. That's 24 except for three minus 16 times three. That's 48 x squared 45 times to that's negative 90 x and in close to 54. So we're gonna set this equal to Darryl and factor are following Ono meal. Okay, so we see that we can practice into we contract out of six and then we have, um two X minus one a two x plus three and Leslie, it's minus one or expert mystery. Okay, so we have three critical points. Um, that's when it's most amount, actually. So, Well, estar en points as well. So stuttering. What? Make it a five. And then we have, um X is equal to negative 3/2 X is equal to 1/2 exit with drink and X equals five. And now let's plug in these X points into our function after backs to see our puppets. Okay, so we see that we get the following outputs for our ex terms. So we see that our absolutes Max, is when X is equal to negative five and our absolute minimum. That's when X is equal to negative 3/2

All right. So here are derivative of our function is X minus one time's experts, too. Time's explains three. Okay. And so first of all, we want to find our critical points said these air X values in the domain of F crime. Where that primacy there. Zero undefined. So we see X is one is going to make a crime. It's your exodus Negative, too. Next is three to have three critical points and then we wanted to get the open intervals where f is increasing or decreasing. So that means we want the intervals. We're F primes, either positive or negative civil client clerk, critical points on the number line and we'LL look att f crime and so to the left of negative to this factor is going to be negative. This factor is going to be negative and this factor is going to be negative. So we have negative making negative. The firm is going to be negative less than negative, too. Betweennegative too. And one will Now this term is going to be positive, but these two are still negative. So it's going to be, ah, one positive into negatives. So that's going to make a net positive and then between one and three. These two terms, we're positive that this was still going to be negative. Negative. And then to the right of three, all three of these terms or community positive. So for F death is decreasing than increasing. Decreasing Increasing. So decreasing. Uh, negative infinity to negative to increasing cremated Teo One increasing from one, two, three in an increasing saying through your insanity. Okay. And then we can see where our local extreme are so at negative too. We're changing from decreasing to increasing. So we have a local men at X equals singers too. We have a local maximum one because we changed from being increasing to decreasing and then another local men that X equals three because we changed from being decreasing but increasing. So if you just want to visualize what's happening or function is behaving like this

Yeah, for this problem we are told in part A to find the critical points of the function F of X equals six X power four minus 16 X squared minus 45 export 16 X cubed minus 45 X squared plus 54 X plus 23 on the closed interval from naked 5 to 5. So first thing that we want to do is take the derivative of that with respect to X. Just going to give us six times four. So that's going to be 24 x cubed minus three times 16 which is going to be 48 X squared. So we have the minus 48 X squared. We have minus 45 times two X. So that's going to be minus 90 X. We have plus 50 for what's that supposed to be minded minus 90 X 54 we want to solve for when that is going to equal zero are going to be equal to zero. Now, one moment. Now that expression can actually be factored as six times x minus three times two X minus one times two X plus three. Which gives us the roots of x equals three, X equals 1/2 And x equals negative 3/2. Yeah, For part being were asked to use a graphing utility to determine whether the critical points correspond to local maxima. Local minima or neither. So at negative 3/2 that's going to be a local minima at X equals one half. That's going to be a local maximum And at x equals three that's going to be a local minimum. And then, lastly, for part C, were asked to find the absolute maximum and minimum values. The absolute minimum is F of three Equals -166 and the absolute maximum is f of negative five Equals 4000, 378.

It's a function Apax calls execute export Begin, Get the critical number Said the first a dysfunction it needs the relative toe Get Didn't believe apple affects our crime. X that equals toe three x skirt my no stole X Now set the bottle A pop remix considers we have You know that is equals toe You export my Mr X after out three x So you have three X that is equal times the quantity X minus four So now we can set the value of X the zeros or this one is X equals zero and access equals toe. For now, we have to maximum relative and minimal relative here. So for Maxima, really relative maximum, uh, get the value off our minimum. So we have to set the Molly off result on derivative extra zero beauty body. So they have fo zero. That's Sequels to 15 on the other. One is apple before. So how part give? That's supposed to 64. My loss. So you have sport sport or 16. 16 times six waas 96. But it's negative. Ninth, six of the last 15. Um then they have seven 29 minus 96. This recalls the negative 17 maximal relative Maxima in which you have 3. 15. Resist the point. I am sir 15. And the relative minimum is, um for negative. 17. So the interval we can use the interval opinion A critical number. So we have negative infinity peril. This is an interval. So that f off prime. But the speaker basis, you say negative one so negative interested They're here. So we have negative three times negative vibe that it's positive team. So now this in trouble is increasing function. When other one she wrote this in terrible can set a value excessive equals to one oh, substitute in the derivative. So we have because so and so you have three times. This is not once or negative 11 that the police Negative five or we have negative one That is positive. One so thes past three I negative three. So it is supposed to a negative nine. So this is a decreasing function. We have another interval from zero in it. So we have terrible. So when we have value off half of prime that it's supposed to X equals five. So we have three times. But that was supposed to 15. We used this substitute off Mexico Spike. So we have five minus four article swan. But this is 15 since pastie. So this is a increasing function at their I have no the increasing function decreasing function of inter balls off dysfunction. And this is the group, as you consider, is an increasing function, decreasing function and increasing function. That's your take.


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