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31,) Absolute maximum value on ( 0, ~ for 12 f(x) 20 3x...

Question

31,) Absolute maximum value on ( 0, ~ for 12 f(x) 20 3x

31,) Absolute maximum value on ( 0, ~ for 12 f(x) 20 3x



Answers

Absolute maximum value on $(0, \infty)$ for $$f(x)=20-3 x-\frac{12}{x}$$

Hello Affects the green function Affects is he calls to 20 minutes three x minus of dwell upon X. We have to find the absolute maximum value on the given interval. 02 infinity, This is opening table and we know that this is a recent function and it is not. It is not defined, that is, and the function and defined at X equals to zero. Okay, because access nanometer of this function. So we will find after sex that did you cost too minus of three minus of mhm man is willing to travel upon access where that will be caused to call upon access square monastery. There still have their sex when, after sex pretty close to zero. Then we will find the critical value. There is four upon three that is X squared. So x will be cause to Yeah, yeah, plus manage to And, uh, there's and the one critical value zero. But at that point, function is undefined and it is not included in the function. So we have only two critical various press managed to. Okay, there's this, uh, minus two. This is a place to message managed to This is flash too. Okay. Sorry. We have only one critical value because minus two is is also not in the integration in trouble. So we have only one. Okay, One critical value that is pressed to. So we will check sign. We will make some charts about it too. So when access less than two, that is suppose one. Then it is posted. Mhm. When I get er than three, then it is. When is when X is greater than two? That is negative. Mm hmm. Okay. Then it is negative. So we can see that it is maximum at extra cost to here it is. Maximum execution too. Exit constitutes is a maxim point. Yeah, So we can ride maximum mad. Yeah. Have accepted answer to that is 20 minutes three and two to minus Taliban too. So that is because to 20 months, six months. Six. That is the cost to eight. So maximum we absolutely have to. That is the cost to it. This is the answer. I hope you and I should thank you

We want to find the absolute minimum and absolute maximum values of the function F of x equal to x cubed minus three X squared minus 12 X plus one on the interval Added to three. Which is a closed interval. So the first thing we got to notice is that F attains that those extreme values that is its absolute minimum and it's absolute maximum values on this interval because interval schools and the function is continuous because simple normal function. So we know that those extreme value exists. But besides that we know that uh those extreme values are images of either the end points of the interval or critical numbers as a function. So we got to start by calculating too Critical numbers of the function for that. We need the first derivative and that is equal to six x Square -6 X -12. And we'll see that this derivative exists for every value X in the interval negative 23. And for that reason the only critical numbers f king half are those files of X. For which the derivative is equal to zero. So yeah. Mhm. And then we get a start by solving the equation After relative equals zero. And that's the same thing as six X squared minus six. X minus 12 equals zero because the relative discretion here and that's the same as six times X square minus x minus two equals zero. And because he six is not no longer we can say this is the same as x square- X -2.0. And we can factor out this polynomial S X minus two times X plus one in that equal zero. And this product series either as a factory zero, so X is two or X is negative one. And these two values in this case are both in the interval negative 23. So they are they're both has to be considered. Then the two critical numbers of F In the interval netted 2, 3 are two and 81. So we got to find the images of these two critical numbers and the images of the end points of the interval. F at let's say negative too is equal to two times negative two cube minus three times Native to square minus 12 times -2 Plus one. And if we calculate all these we get negative three, that's the image of the left hand point negative two. Now the image of the writing 20.3 is to attend three Cube -3 times three square mm hmm minus 12 times three Plus one. And that give us -8. And now we have a late at the critical numbers so F two, he is two times two Q minus three times to square minus 12 times two plus one. There is negative 19. And finally f at the other critical number 91 is two times negative one cubed minus three times negative one square minus 12 times negative one plus one. And that give us and eight. Mhm. So the largest value of these four is eight with the absolute maximum of f over the interval. Native to three and the smallest Of the virus is -19. Okay. So this is -19. And see absolute minimum of the function over the interval. So we got that answer then if has an absolute minimum value 80 negative 19. Which accor's at uh to which is one of the critical numbers at the critical number. Okay, mm It's equal to and f has an absolute maximum value, maximum value we said eight. And that value of course At -1 which in fact is the other critical number at the critical number Ux x equals 91. So this is an example where the the two extreme values of the function happened or a core at the critical numbers inside the close interval. So this is the final answer and then propping all with it here. First thing we noticed that the extreme values of the function exists because the function is continuous and defined on a close interval. And we know those extreme values are images of either the end points of the interval or critical numbers of the function we find the derivative and because that derivative exists at every point or a number in the close interval. So the only critical numbers of the functions are those surveyed for which the relative is zero. We solve that equation derivative of ethical zero. We stick with the solutions that are inside the close interval In this case both of them are so we have a lady function at two critical numbers, we found, and the end points and the largest Value of these four values is the absolute maximum of the function over the interval on the smallest of the values is the absolute minimum value of the function on the into. So this is a final answer to this problem yeah.

Go to farm. Their stupid Maxwell minimum affects is given. Do you excuse Monastery at the square two x Q minus three. Access square minus 12 X plus one minus 12 X plus one. The interval is managed to common trees. Okay, so if you can find the address X that is close to six X squared minus six X minus 12 because 20 x squared minus X minus two because 20 so it will be a prescribed minus two X plus X minus two. It calls to zero so we can diet X minus two explicit one it calls to zero. So at the critical point will be two comma minus one. The function value F off my husband and help of two. Will we mhm Mm. It will be f f minus when it comes to eight and effort to question 1009 and now they're in trouble has given minus two comma three. So we will also find the value function value of the function and the empires. So the function of the value of the function had managed to managed customer history, and I have three custom minuses so we can see that the absolute. Okay, the absolute maximum. Therefore manage when it comes to yet. Yeah, and absolutely minimum value. Mhm x. Of course to that is minus 19. Mhm. I hope you understood. Yeah. Thank you. Yeah.

No we have to find the absolute maximum value and after criminal for the given function effects that energy costs too. Excuse me. This creates a square one is one okay. On the enormous street too. It is a close in trouble monastery to to. So first we'll find the basics to find the kid Columbia. So it will because of three X squared plus six X To fund the curriculum. But it does actually because 20. So it is when we cost you three we will take common from here. Three X. It's X plus two. Express to mhm. Any cost in a row. So accept quest to zero and acceptance to manage to. So these are the tube interval two critical number that belongs to the yeah closing double monastery into me. So now we will find the function value at and point that is a fault monastery and therefore he managed to could and technical McMahon points for europe and ethel's Who? Yeah. Okay so 4 -3 will be monastery Q plus three In two months. Three Holy Square one is one. Okay. It will have managed to holding there's three into one is two whole square minus one. So it will be zero plus zero minus one. So it will be Holding Plus train to two whole square minus or what? Right. Okay. So it will be cost to minus of 27 plus 27 -1. She will read -1. Okay. Yeah managed to. Hopefully that is minus eight. Mr one is one. That will be close to three. So you didn't request to -1 And that will be caused to eight plus. Well one is 1. So it will be close to 19. Yes. Okay. So from this value we can say we can see that absolute number lose management. Yeah value In the -1. Because it has a lower will f of zero. Never Minister has a lower will and absolute Mhm. Yes, maximum value it comes to 19. So these are dancing right? I hope you initial. Thank you.


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