find-the-ordered-pair-x-y-on-the-graph-of-yx3-3-x2-for-which-fprime-primex0-this-is-called-an-inflec

Question

Find the ordered pair $(x, y)$ on the graph of $y=x^{3}-3 x^{2}$ for which $f^{\prime \prime}(x)=0 .$ (This is called an inflection point.)
Show that $\frac{d^{6}\left(x^{6}ight)}{d x^{6}}=6 ! \quad(6 !=6 	imes 5 	imes 4 	imes 3 	imes 2 	imes 1).$

Bryan Lynn · Accepted Answer

Okay, So here were given function up, which is gonna be equal to X cube, my ass 18 x squared, minus 10 x plus six. And so we need to find the inflection point and, Graf the function. And so the inflection points occur when the second derivative is equal to zero, we&#x27;re gonna find the second derivative, and we&#x27;ll take the first derivative. Hey. And so the derivative of X cubed will be three x squared. The derivative of minus 18. X squared will be minus 36 x and then we&#x27;ll get a minus 10. And then that, plus sex alternative zero. And now to take thes second derivative taking the derivative of the first primitive, we get six x minus 36. Okay, and this is actually equal to six times X minus six. Okay, so that&#x27;s gonna be equal to zero. So the second grader, the axe is equal to zero when x is equal to six. So that&#x27;s where our inflection point will occur. Okay. And if we go ahead and graff this recommend using ah software to get something a little more accurate, but and of course I work. So what? Thio. He says I thought we get a function that looks something like that. Okay. In the Net, X equals six. Here. You would have your inflection point and these air not quite to scale. This is going to be while six is here. Uh, this would be about negative 500.

Zachary Watson · Answer

finally, the important values for dysfunction. I&#x27;m gonna first take the first derivative of this thing. So at problem of X, and in order to do so, I&#x27;m going to use the product roll. So first I&#x27;m gonna drive X my six, which would give it one more inspired by Expo Six or squared something just be left with this. That&#x27;s gonna be added to derivative of experts. Six square is going to explode. Six powers can produce to wine, and I&#x27;m gonna multiply by X minus six. That&#x27;s not equal to zero, Right? So one thing I see here can do is have to take out one expert six. That&#x27;s gonna leave your experts six because two x my six. Could you call zero Clearly experts sixes equals zero. X must equal negative six. That&#x27;s one value right there. And then I&#x27;m going to expand everything else inside. So X plus six close to X minus toils zero three X minus six equals 03 X secret six x must equal to That&#x27;s gonna be our to critical points right there. So I&#x27;m gonna take one more derivative double crowd lex to find inflection points. So I&#x27;m gonna get to next six to first power plus artist of the product rules. So to takes, my six course to express six from equal deserve. So next I can cancel the twos, right? That&#x27;s gonna leave him with a X plus six plus x. My six plus X plus six equals 0 to 36. Is there going to cancel out? I&#x27;m gonna get three. X is equal to Well, Greeks for six is equal to zero. Three x is equal to negative. Six actually give equal to negative two. So he&#x27;s negative. Two is gonna be our inflection point. Okay, so next off, let&#x27;s look for some X and y into sips. So let&#x27;s say X equals zero That&#x27;s going to give me, uh, negative six plus six squared and that&#x27;s going to give him you. When I worked out to, uh, negative to 16 so they zero mega to 16. That&#x27;s gonna be our y intercept and endless check for way. We said at f of X equals zero. So clearly only do that CR X is equal to six we got earlier, and then we have the square can cancel out, and we get X is equal to negative six. So that&#x27;s two war values are there. 60 60 Okay. And so let&#x27;s go ahead and label list everything out. So we have negative six. And when we have negative six, we said X equal to zero. So that&#x27;s gonna be best for me. Accident. Use up. Then we have a negative too. Of a next legal negative. Two f of X is negative. 1 28 It&#x27;s an inflection point, right? Okay. And then we also have ex equaling two. When X equals two, we have negative 2 56 and that is going to be was that is, uh I forgot to put a number line here, so it&#x27;s worked out. All right, quick. So if we put a number on her here for negative six and two on and try and see what f of X if f f prime of X is positive or negative in the function in the values in between and two left and right. So if we choose valued us less and negative six, we&#x27;re gonna get a positive number for F prime of X. We need that function is increasing at that at that interval. If we choose a number between negative six and two. We&#x27;re gonna get a negative number mean that function decreasing. And then we pick something greater than to function. F Prime of X is gonna be positive, meaning that under functions increasing. So we can clearly see here that negative six is also a max. And that, too, is a May. Okay? And so, dang, what? We have sex. We have also we have 60 All right? Yeah. And then 60 is whole, actually, let&#x27;s let&#x27;s go back. First, we have zero. An idea to 16 and you don&#x27;t get to 16 is a Y intercept. Dan, we have 60 which is an ex intercept. Okay, so with all this, all these values here was trying Craft that out. This is just gonna be a rough sketch. We have some very big numbers in here which make it a little difficult. So try the best you can do. Like so let&#x27;s say that on the intervals are, uh, say 50 50. Take it for a steal. 2 50 These intervals here, this is gonna be one B r. Okay, so it started Negative. 60 Right. Then we have No, you too. Maybe 1 20 A Someone there. Well, we have 0 to 16. Some right there. We have to get to 56. And then we also have 60 go. Okay, so the function is basically no exulting. You must double, but because in Tuesday Inflection point. Yeah, I know that looks a little rough, but he&#x27;s gonna look something like that. That&#x27;s clear. Gonna be a little smoother, but that should be with a graphic site. We&#x27;re off sketch.

Zachary Watson · Answer

Okay, So to find out the special values for dysfunction first so we can do is we take that first derivative so f from vets that&#x27;s going to be equal to six times 2 12 extra moneys. 12 extra third. Nothing the equal to zero we can send equal is there&#x27;s we could find these critical values. I went there, and I see here that we can pull out is a 12 x to third. Just gonna leave us with, uh, X squared minus one song is equal to zero. So clearly it told Exit third is equal to zero. That means X plus B equals zero. Then X squared minus one equals zero. We would want other side sex. Critic was one expert secret of positive and negative one. So that street critical values there that we found just put a number line up right quick zero negative one and one. And it if we were to pick a number less than negative one between negative infinity negative on and put it into F of X, we get a largely negative number showing that for FX in that integral is decreasing. And then if we chose a number between negative on a zero uploaded to F problem of eggs were going to get a positive number showing that in that interval is increasing, then between zero and one that we will get a negative number for plugging my number in between that for f prime of X, so decreasing again and then anything greater than when we&#x27;re going to get a positive number for convicts. So then defensive eat, increasing so clearly big own because of this craft, Would you see that negative one of one our minimum values while zero is an axle. Now let&#x27;s look at the second review a double prime of X. This will show us our inflection points. Okay, so we&#x27;re gonna most by five times told we&#x27;re going to get 60 x to the fore minus 36 x squared were set that equal to zero to find the infection quakes. Okay, I can carry out a 12 x squared out of that and we&#x27;ll be left with five x squared. Minus three said they could zero. So 12 X squared is equal to zero. Excellent Speaker zero and then five X squared. Mostly quick. Three X squared would be good if 3/5 accidentally Good applause of *** scared of Greek over five So that&#x27;s on 23 inflection points right there. Now let&#x27;s look it to see if we said every set X equaliser So go zero is going todo was your my 00 So clearly 00 is gonna be We said What? X equals zero system your white Why interested one x and y interested And then the sea. When y is equal to zero, we&#x27;re gonna have to x to six my street x fourth and so something else you could take out is ah X squared. No, sorry. Exit before. So for ex support equal to zero, clues are zero. Must be the answered wearily Fonda in the previous step. So we&#x27;re just gonna cancel that out, So I&#x27;m going to disappear. Don&#x27;t say two X squared minus. Wasn&#x27;t he could three when you moved at, uh, three over to the side. So x squares when we goto 3/2 X is going to put a positive negative screw 3/2. And that&#x27;s gonna be for while Isaac Reserve. So let&#x27;s list everything we&#x27;ve got. So we have 00 That is, uh, a local max. It&#x27;s also our Brazilian X and Y intercept, and then is also named collection. Then let&#x27;s look at when you put in negative one and when you put it in one so X equals negative one. Uh, fx also gonna be good negatively when x is equal and one is also going to be able to negative one. And these are actually gonna be absolute minimums for function. And then we have square with the revive Enigma Square. Do you have I? And when we put that in, we should see the for both of these. They&#x27;re gonna be equal to negative six for eight. And these are gonna be inflection points on efforts, girl 3/2 and negative script 3/2. Y was equal to zero. So these are ex intercepts. So why don&#x27;t we put all these into a graph? Just a rough sketch, Right? We have. Ah, say Syria. Zero things are going. Thank you, boy. One negative one. Okay, we&#x27;re gonna have positive native script. You find negative 6.8. So to me, be somewhere around here, and then we&#x27;re also gonna have positively sphere screen would be over 20 So I wonder you&#x27;re on, like here. So? So the graph coming is gonna look something like this. Why? Might look a little weird. It&#x27;s just that the graph flattens out here between positive and negative script. You were to those values and they&#x27;re very close together. Your home 20 But that&#x27;s what the graph should look like.

Eric Mockensturm · Answer

in this case were asked to PLA y equals X cubed minus three X squared plus six X. The first derivative is three X squared minus six exports six second derivative ist six X minus six. We can look for transition points and we have an inflection point when X equals one. So why double prime at X equals 10? This has complex roots, so there are no no extreme. So there&#x27;s no maximum or minimum in this curve we can look at Obviously it X equals zero. Why is zero and X equals? X equals one? You can put that in and we get Why is four and so also notice that we could factor out an axe here, Um, so we get a X squared minus three x plus six. But again, that has complex roots also. So there are no other zeroes. Besides what at X equals zero fighting that we have, um, at X equals zero. We have why zero X equals one. Why is four knows? We got very large scales here, and so we have negative curvature, positive slope all the way up to this point at one. And then we have positive, um, negative, positive curvature, positive slope for the rest of the way.

Jennifer Fowler · Answer

consider the function y equals negative to excuse. Plus six Ex queen minus first The white prime is negative. Six X squared plus 12. Okay, so this is how you take it? Negative six negative. Six packs. I have minus two. So we have critical numbers. Numbers that equals zero. And okay, Emma will probably occur and then was fun. Why? Devil? Prime. So that&#x27;s negative. 12. Thanks. Trust will and noticed that this is equal to zero x one. So there&#x27;s a possible inflection point. Ah, people. It was finally a function values associated with this so we can go ahead and through all this grab. So recall that we have y equals negative to excuse. Plus six X squared minus three. So far, luck and zero. I get negative. Right? So we know that the point. The road negative stories on the ground. If I plug in one, I get it Negative too. Plus minus three. Which one? So I know that the 0.11 is on the ground. And if I plug in to I have negative 16 What? 24 minus three? Nothing. Five way. Know that we have the point too. Next. Okay, so We have to go. 12345 down. Three. Okay, so we have the point zero negative. Three. The 0.11 And we have the point, too. And the grass looks something like Okay, so there is. Remember, this is our inflection point. There&#x27;s a local man. There&#x27;s a local map and there&#x27;s an inflection point at next.

Problem

Solve the given problems by finding the appropria…

Problem 40 Easy Difficulty

Find the ordered pair $(x, y)$ on the graph of $y=x^{3}-3 x^{2}$ for which $f^{\prime \prime}(x)=0 .$ (This is called an inflection point.)
Show that $\frac{d^{6}\left(x^{6}\right)}{d x^{6}}=6 ! \quad(6 !=6 \times 5 \times 4 \times 3 \times 2 \times 1).$

Answer

Related Courses

Basic Technical Mathematics with Calculus

Related Topics

Discussion

Top Calculus 1 / AB Educators

Anna Marie V.

Heather Z.

Kristen K.

Joseph L.

Calculus 1 / AB Courses

Precalculus Review - Intro

Functions on the Real Line - Overview

Recommended Videos

Watch More Solved Questions in Chapter 23

Video Transcript

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Anna Marie V.

Heather Z.

Kristen K.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

Basic Technical Mathematics with Calculus

Anna Marie V.

Heather Z.

Kristen K.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Anna Marie V.

Heather Z.

Kristen K.

Joseph L.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus