Problem 20 Medium Difficulty

Use a computer algebra system to graph $ f $ and to find $ f' $ and $ f" $. Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of $ f $.

$ f(x) = x - \tan^{-1} (x^2) $

Answer

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Video Transcript

we want to use some kind of cast software to graft. Apple X is equal to X minus Katja inverse of X squared and find the first and second rib tips using the software. Then we want to graft the derivatives to estimate the intervals of increase, decrease con cavity, find any extreme values and points out inflection. So I would have had it already found the first derivative and grafted alongside our original function here on this first page. And to get our viewing window or our original function, all I did was zoom out far enough to see a We had any kind of interesting activity going anywhere and outside Really X is equal to two. That really didn't seem to be much. So this is the graph for FX that I'm going to leave us with. Then what? I went ahead and grab the first derivative here. I did pretty much the same thing and outside excessively negative you do again. We really don't have much information, but remember for the derivative, the only thing we really care about since we're looking for um, extreme values and where the function is increasing and decreasing our Exeter's so I just made sure I included all of our ex intercepts into this. And so the two exits upset found where? 0.5, or and at one? Let's go ahead and find where this is increasing or decreasing. So remember, defied with the functions increasing, that is where F prime of X is strictly larger than zero. So coming from negative infinity, the function is larger than zero and it will be larger than zero. We had our first intercept into that 0.5 warp and then from one to infinity, it will also be, uh, positive. And then our function is decreasing where f private, that's strictly less than zero. And this only occurs between 0.5 or four and one. Now, what we can do is determine if these two ex intercepts that we have our first resident we're going to be maximums or minimums for our original. So to the left of 0.544 since prime of X posit, that means we're increasing into that point. And then on the other side of it, it's negative. So we know we're decreasing. So that tells us by the first trip to the test that this here will be a local mats and for one well to the left of it. It's negative, so it will be decreasing. And to the right of it, it's positive. So it's increasing. So that tells us we will have a local Men at X is equal to one. So that's everything we can really get from the first ribbit. Now, for the 2nd 2 resident, I want to have it did the same thing, uh, insured. I had all my ex intercepts included. And those Exeter such for the second year of them into the negative 0.76 and 0.76 And we'll just do the same thing we did before defying contact of the of being up and down. So it is Kong Kate up when f double prime of ETS is strictly larger than zero. So starting all the way at negative infinity, it will be calm paid up until we had our first intercept negative 0.76 And then when we had 0.76 to infinity and it'll be conch aid down on the rest of our interval. So con cake down is one app double pond Alexis Tripolis. And this is going to be a negative 0.76 to 0.7 six. Now, to find points and inflection, we need to see that there is a change in con cavity. So to the right of our first X intercept, it is going to be Khan caged up and to the left of it is going to be calm cave down. So we do have a change of inflection there. So that will be a point of buckshot. And for other X intercept 0.76 it is calm. Cave down, toe left, calm Cabe up to the right. So we also have another change of inflection there. So we found our points of reflection. We found intervals where the function is increasing, decreasing call cavity and we found our extreme