Use the guidelines of this section to sketch the curve.
$ y = 2x - \tan x $, $ -\pi /2 < x < \pi /2 $
So, given this function and this domain, we know that the intercepts of the easy 100 and they want it a little harder to see that's within the Stone Main is negative. 1.20 and, um, positive 1.20 So the symmetry is odd. The assassin totes because it's Do you see a tangent here? It's a repo over to end supply over to them, and now we're going to see what the increasing and decreasing intervals so we have to solve for y prime. So to do that here, why prime equals to minus sequined X seeking squared eggs equal zero we're setting. It's easier to find the critical point. So X equals pi over four um, pira for just the one that's in our domain. But it's obviously a trig functions, so it's gonna, you know, have multiple of values across different periods. But this is just in our domain. And so let's do a first riveted test for this. So it's every pi over four. So we're gonna have a negative play over four in addition to the positive one and substituting values and our first derivative, um, that are smaller than negative fire before we'll see that it's decreasing Betweennegative piper for empire before his increasing and anything after pi over four is decreasing again. All right. And if we want to sew for if of pi over four to find, um, the Y value for our well here, it's negative to positive, so decreasing to increasing. So this is the minimum, and here is increasing to decreasing, so this is a maximum. So if you want to find the maximum, for example, the Y value would be pie over to minus one, which is approximately 0.57 All right, so this is the first derivative test to show increasing and decreasing. All right, if that's our first derivative or second derivative, is going to be negative too. She can't squared. Thanks. Times tangent six. All right. And if we saw, um, if he said equal to zero, you see that X is equal to every pie n and is an integer and so pies Obviously bigger than what's in our domain. We don't hit a pie, you know? It's pies out here, pies out here. We're not within that. So, um but we do see that since It's sometimes an energy we can do. We can do, um, time zero and get zero, which is our intercept. So we can test for zero on our second orbit of test for Con Cavaney. This is a double prime crew zero, which again is our intercept. This'd inter an interval. So if we test for a zero, anything less than is gonna be positive. Anything greater than is going to be negative, which really means Khan gave up and Khan came down and says the switches from positive to negative. It switches signs. It's an inflection point at zero. And now we can graft. The first thing we can grab is the S on too, which is pi over to end. So it's Let's label that negative pi over too. And positive play over too. All right, so what's labeled that in green? So these are our ass. Mentos. You're gonna be vertical. So negative and positive player ever to we have some toast. And now we have a minimum value here at negative pi over four. So if this is pi over to in the middle is gonna be a negative pi over four. So say it's summer down here. And I know it's somewhere down here because we do know the why value for the max, which is so pi over four and the Y values about 5.7. So if this is, let's say say, this is one negative one, and this is one. So the negative 5.7 would be somewhere here. And since it's odd they're gonna be like that because the symmetry All right, so now we have our men and Max and we have the inflection point and intercept that 00 And we have our, um, other intercepts somewhere here is gonna do a rough sketch. Um, so we have decreasing from until we hit, our minimum value is gonna be decreasing. So it's gonna look like decreasing, hit the axis, decreasing and then hit our minimum, and then it's gonna increase. And then it's gonna decrease again. And you won't touch her ass until but I'll get really close to it. And let's check for con cavities. So, yes, Khan gave up. Then we have the inflection point and then conk you down