Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).
$ f(x) = \cos t $, $ -3 \pi /2 \leqslant t \leqslant 3\pi /2 $
Absolute and local minimum value $-1$ at $x=-\pi$ and at $x=\pi$. Absolute and local maximum value $1$ at $x=0$.
here we are going to sketch the graph of the function. Go sign of T. For the argument in the interval from negative three by half. Three by half. Including both endpoints with that craft. We're going to find the absolute and local maximum and minimum values of the function. So we're gonna use this system here. This coordinate system. We're going to do more or less the graph of the coastline function. We know to go send a serious one so we can have something like this. Okay let's go sign the here's a little bit better than that. And then we have this far here and this bar here more or less. And here we have. The value is included because we have included both in points. The value of negative three by half and three by half zero. Mhm. Sorry. Here see your rock. And this value here is one and this value here We know it's -1. That is at this point. The one which is the same as this point here. And as we can see we have here all the elements we can have in terms of extreme extreme A. So we have first we have the absolute maximum value of the function is one, attain it zero. But it's also local because if you look any interval Near zero or contain zero. And look at the graph only there again the value you want to see how his value. So F. Has uh an absolute and local maximum and local maximum value. Yeah one at X equals zero. So that's the first thing then we have that The minimum value of the function is -1. But it is attained at two points in this case. Let's see that these values here here we have by half here we have negative by half. And this number here is by and this number here where we have negative one is by its negative. So this here is negative. That is plus three by half for you. And here is negative three x half. Which is the man we are considering this case. So we attain the the minimum value negative one at two points at by an ad 90 by So if and they are also local because if we take any interval containing the These values the value -1 is the smallest image possible. So f has an absolute minimum value which is also local -1 at the point. Or the numbers X equals negative pi and X equal by And in fact it because he was the function at. Find it under real numbers there will be infinitely many points. Our values where we attain the minimum value in 81. There will also be infinitely many values numbers where we attain the maximum absolute maximum body wants. But in this case we have to train the function to this interval And over there we have this situation if F has an absolutely local maximum value one at zero here. This point here, N F has an absolute and local minimum value in anti juan At two points here and here. So that's the behavior of the go sign of T for tea in the interview. From negative three by half and three by half, including both endpoints.