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Problem 68 Hard Difficulty

(a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places.
(b) Use calculus to find the exact maximum and minimum values.

$ f(x) = x - 2\cos x $, $ -2 \leqslant x \leqslant 0 $

Answer

(a) SEE GRAPH.
The absolute minimum value of $f(x)$ is $\approx-2.26$.
The absolute maximum value of $f(x)$ is $\approx-1.17$ ;
(b) Absolute minimum value is $-\frac{\pi}{6}-\sqrt{3}\approx-2.2556495832$ which occurs at $x=-\frac{\pi}{6}$.
Absolute maximum value is $-2-2 \cos (2)\approx-1.1677063269$ which occurs at $x=-2$.

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Video Transcript

using the graph we want to estimate the absolute maximum and minimum values of function keeping or define as x minus to co sign of eggs. For eggs in the closed intervals for negative to zero we're gonna use two decimal places to do that estimations. And in previous calculus to find the exact minimum and maximum values of the function. So we have here a graph of f of x minus to co sign of eggs. As we can see on the close interval from negative to to cyril. Um This function is negative and has this shape here, this blue line as we can see we have uh minimum value, relative minimum. In fact an absolute minimum meant very close to disappoint tape highlighted here and we have a maximum value of the left hand point negative too. And this value is around somewhere here at that point negative 0.5 something. So we can say that if we take a close look at this region here on the graph, we have this graph here. And as we can see, we have a absolute minimum of about negative 2.26. So or negative to 25 if you want. So we have here absolute me and your mom About -2.26. And these corresponds to this region over here. Now we know that the maximum absolute maximum value of the function is attained at the left point of the group. That is a negative tomb and a close up of that region is given here. And as you can see, the maximum value is around this. By the here, which is About negative 1.16 so or negative 1.17. The absolute maximum About -1, 17. Yeah. So graphically we have seen these values and now we want is calculus to determine exactly reach these points are So it's our eight. Now I'm 40 these capitalists and we're going to apply the fact that this function F is continuous. Remember the formulas X -2 cold sine of X. And since co sign of X and X are continuous function, these Expression X one is to cosine of X is also a continuous function. And it's defined in this case in a close interval negative 20. You know that the continuous function and defining a close interview must attain the extreme values. There is a mass absolute maximum and an excellent name on that interval. But we know more than that. We know that those extremes extreme values must be obtained either at critical numbers of the function inside that interval or at the end point of interval. So we got to find the critical numbers that are inside this interval and with that and the endpoints we calculate all the images of those points and the greatest of the images is the absolute maximum and the lowest of the images are the absolute minimum on that. Close the interval. So we're going to calculate force very negative of F which we now is able to one minus or floss to sign of X. Yeah. And so the first derivative is zero. Even only if one plus two sign of eggs, it's zero. And is equivalent to saying that sign of eggs Is equal to -1 health. And if we look at the function sign of eggs on the internet, native to zero, we have here let's say we are zero. So we have this we are here at negative by house, we have negative one And then we have zero value at native by but native by is found 83 points 15. And then the answer to negative by half is about 91.57. So you know negative tooth around here it's pretty better maybe we have negative to hear around here. And so the graph is more or less this I'm talking about sine of X. And so If we are looking for developers of X for which sign is negative 1/2 for eggs On the Internet, addict to zero We have the negative 1/2 is about here. And so we are Looking for this intersection here where sign equals -1 health. But what is important is That is only one solution. There is only one solution to this equation. four eggs on this interval This interval native to zero. Yeah. Now we know that that Angle X. is negative by six. So the only can say that the only solution two Sine of x equals negative 1/2. Or eggs on the interval negative to zero is X equal negative by health. That is because we know sign up by by six payment Because we know that sign of by six is 1/2 and sign is an odd function. So sign of negative 56 is equal to negative sign of basics. That is negative one house. Yeah. So we have this Here is pretty better six. And this value is about if we want to know negative 052 3598 7756. So it's about values. And we can see here this graph it's about that We found here with two decim is around negatives 52. But we have found exactly or more precisely these negative 0.5235987756. Or mathematically is negative by over six. That's where the absolute minimum. Uh That is uh That is a critical point or critical number of the function on the interval negative to zero. That's zero. That is because the derivative zero only at that value. The series of derivative art critical numbers of the function. So the function has only one. Critical Yeah. Uh number mm X equal 95 6 in -2. We also verify that this value is inside this interval. That is clear that that's the case. And we also had like the fact that the fact that this narrative exists for any eggs. That is dysfunction is well defined at any value of X. Any real number X. So that's why we look for critical numbers to this function only solving this equation here. And that is because they cannot be in this case. Critical points, correspondent to uh huh derivative non existent at some point. Because the relative in fact exist at all points in the real numbers and very in particular in this interval. Okay, we have this value here and without that this can be a number is one. Here can be a number where we have extreme value. We got to a way to function at that point. We have that F negative 0.6 is equal to negative 0.6 minus two. Cold sign of negative by six, which is the same as negative 56 minus two. And co sign of narrative by six is the same as Co signer by six. That is square 12, 3/2. We simplify this factors to here and this give us that F at negative 5 6 Is equal to -56 Minour score to three, which is about negative 2.25 56 49 58 32. And that's important to have this value because we had to compare all the images at the critical points and at the end points of the sub interval to know which corresponds to extreme values. So we have that and then at the end points That is F at -2, which is the left hand point is equal to negative two minus to co sign of negative too. But that's the same as negative two minus to co sign of two. And here we get these calculators about negative one 16,770. Uh 632 69. This is another result we got to take into account. They mentioned the left and point of this of interval of the interval. Sorry. And now we got to calculate f at zero and that is zero minus Two times Gulf sign of Syrup was in a serious once or get 100 -2 is -2. And this is the value. So we have the images of the end points of the of the close interval is to hear and we have the image that the only critical point inside the interval native to zero which is here from these values. We after the greatest of the values Or the largest value is -2. Um sorry negative 1.16 16 77063. So this is the maximum value And the minimum value is -2. So more sorry negative 2.25 because needs to be integrated. So we have to extreme values here. Extreme values now. All right, absolute maximum negative Let's say f at -2 which is native to -2. Coz I have two which is about 91.167706 three 269. And we have an absolute minimum and this stream value here is attained or of course at native to, it is clear because I'm putting here is f at native to and the absolute minimum of course at Uh the critical point active 5/6 And that value is exactly equal to 25 6 Minour exc words of three, Which is about -2 um 25 25-5649 5649 5832. So those are the answers of the eastern values. So the absolute maximum value of the function Just around 81.1677 is It occurs at -2, which is the left hand point of this interval And the absolute minimum is around negative 2.2556. And it of course at the critical point negative by over six, which is in fact the only critical point of the function Look located art or in the interval an attitude zero. And this corresponds to what we saw graphically because we have the absolute maximums 91.16 was around native to in fact was exactly at -2. As we can see here. This is on the line. He says that we have the image of negative to the image is about 91.17 which is in fact negative 1.1677063269. And the absolute minimum as we saw here in this graph is around -2.26. And of course at negative 0.52. But in fact it of course exactly at negative by over six, which is around negative 0.52 35987756. And the value that is the absolute minimum, in fact, is around negative 2.25 56495832, which mathematically is equal to net it by over six minus score to three. So we have, you can see that mathematically that is using calculus. We can find very precise results, but graphically we can have a rough idea of what these values are and this is the finance solution of the problem.

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