00:01
To find the local and global extrema we'll find the critical points.
00:05
Okay these are the points where the derivative of f is equal to zero.
00:11
So let's start by finding the derivative of f.
00:13
This is equal to negative sine x minus one half.
00:19
So i'll set this equal to zero and solve for x.
00:25
So we get that sine x is equal to negative one half.
00:34
Okay so i know that sine of seven pi over six is equal to negative one half.
00:41
So we have x is equal to seven pi divided by six.
00:47
Okay and also sine of 11 pi over six is equal to negative one half.
00:55
So we have x equals 11 pi over six and sine is periodic as a periodic function with period two pi.
01:06
So we can add integer multiples of two pi.
01:12
Okay and still get negative one half.
01:16
But note that due to our interval i can only have n equal to one or zero.
01:27
Okay because if it were negative one or smaller negative two negative three etc it would be outside it would be to the left of this interval and if it were two three or larger it would be uh larger than four pi.
01:44
Okay so it'd be outside the interval.
01:48
So then the critical points are x equals seven pi over six.
01:54
Actually i'll put this in a table.
01:57
Okay so we have seven pi over six, 11 pi over six, seven pi over six plus two pi.
02:10
Okay so that's uh plus 12 pi over six.
02:16
So that that's 19 pi over six total.
02:20
And then we have 11 pi over six plus two pi.
02:25
So that's 11 pi plus 12 pi is 23 pi over six.
02:36
Okay and in this table i'm also going to include the end points of the boundaries.
02:41
So that's zero and four pi.
02:45
Okay and now i'm going to calculate the function f of x at these points.
02:50
So this is cosine x minus one half x.
02:54
Okay so we'll calculate the values and we'll use them to find the local global extrema.
03:08
Okay so first we have cosine of seven pi over six which is negative root three over two and then we have minus uh seven pi over 12.
03:22
This is approximately negative 2 .70.
03:29
And then we have cosine of 11 pi over six which is positive root three over two and we have minus 11 pi divided by 12.
03:42
And this is approximately negative 2 .01.
03:46
And cosine of 19 pi over six is negative root three over two.
03:53
And then we have minus 19 pi over 12 which is approximately negative 5 .84.
04:06
And cosine of 23 pi over six is root three over two and we have minus 23 pi over 12.
04:18
This is about negative 5 .16.
04:24
And then cosine of zero is one.
04:27
Okay and then we have minus one half times zero which is minus zero.
04:32
Cosine of four pi is also one.
04:37
And then we have minus uh half of four pi which is two pi.
04:43
And this is approximately negative 5 .28...