verify-that-the-function-satisfies-the-three-hypotheses-of-rolles-theorem-on-the-given-interval-th-5

Question

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers $c$ that satisfy the conclusion of Rolle's Theorem.
$f(x)=5-12 x+3 x^{2}, \quad[1,3]$

Brenda Knauber · Accepted Answer

the function. Fearful Wrecks the Eagles five minus 12 X plus three x squared on interval from X equals +12 X equals degree. So this was the first to have about this use of rules there, Um, because it is a quadratic function are second order pulling over your function. Eso By that functional definition, it is both continuous and differential on the required intervals. So now we need to check for the third hypothesis of Rose therm, which is that the function is of equal value at the end points of integral. So first truck, the function at X equals one. We have fired more honest toil times one post three times ones squared, which is equal to minus four. Um, a function at X equals three is five minus 12 times three plus three times three squared, which is he also equal to minus four. So, no, we should be able to find at least one value between one in three. We&#x27;re most slope at that point. C is zero. That is, we need to be able to find at least one value where the salt might change from positive to negative or vice versa, or if the function happened to be constant for the entire range, which will not be the case here. Then it would be the entire inch, but we&#x27;re here to find that out. So let&#x27;s take the derivative of the function every edible, minus 12 plus three times two x. Let&#x27;s set that equal to zero. So then minus told host six X Eagles here, move over. A 12 do right, right. Six were you result in having a see value of two, which

Fahad Paryani · Answer

Okay, So this questions asking us to verify the front that the function after practical to x square manage for class five satisfies two three hypothesis of roll student on the given interval Andan We have to find the number of sees that satisfies the conclusions of roast him. So here we know that after Max tickles too X square minus four plus five. So just by looking at this function, we know it is a poly normal because it has a a function of degree greater than one So here too So we know this is appalling amore and all polynomial czar continuous and they&#x27;re also all defensible. That is sick That is the property of all polynomial. So we know off the back that thiss thes two conditions continuous and defense ability are are given of upon your meal And if you actually plug in the values of negative ones So if you plug in aa two of negative one squared minus four times negative one plus five, you get yesterday get eleven And if you do the same thing for three so now you plug in three the two times to re squared minus four times three plus five. It also gives you eleven. So we know that half of negative one and efforts three are also satisfied is also a truth statement. So now that we satisfied all the three conditions required for a roasted um, we can go ahead and find the numbers of sees that satisfies the astral conclusion. And so the recall the conclusion of this serum is that they that we know that there will be a number see, in the interval Judge that the director of the derivative at point C is equal to zero. So we have a horizontal line and we&#x27;ll get you salt for this in this case. So if we take the duty of effort back so the derivative of F Prime I mean F Prime Vex, which is just a derivative. We can find that the derivative of effort back is just for eggs minus for and all they&#x27;re doing is looking for where this function in the Cordillera. So all we do is set this function equal to zero. Then we just bring the fourth to the other side. So we get four x equal four on DH X equals one in self Rex and This is also our number, See, because if you plug in this value one, the four times one is one I&#x27;m in for ten points. Four minus four zero. So that is the sea that satisfied the conclusion? Of course, dear. Hey, there&#x27;s our answer.

Brenda Knauber · Answer

the function of forex people to your X cubed minus X squared, right, A six x plus two on the interval from X equals zero to X equals three. So, as faras the first year hypotheses use our earl staring because it&#x27;s 1/3 order polynomial function our which is continuous indefensible on for all values of X. So now we need to check the third hypothesis of rules through him. What requires that the function has equal value expressed endpoints. So it&#x27;s begin with hope X equals zero, which is your cute Once you&#x27;re squared six times yours equalled poster is equal to two on, uh, actually close very. Which is he called a three cubed lannister squared six times, three opposed to, which also happens to be equal to two. So then those function on this interval satisfies all three have prophecies of rose there. Um, so then they should be at least one value of X. We will see that satisfies, uh, but is has a slope of zero between X equals zero and X equals three. That is, we&#x27;re looking for a crime of Sea Eagle zero. So let&#x27;s take the derivative of our function. Saranda but three X squared minus two X minus six Eagle is here. I can&#x27;t factor this by looking at it. So let&#x27;s use the quadratic equation 300 books to plus or minus the square root. Uh, two squared minus four times, three times my six whole over two times. Three. The sons of being spirit 76. Well, it&#x27;s keeping us two times. Two. Where tons of being two tone swear word of 19 years old and one poster Minuses. Word of 19/3, but positive choice will be between X equals zero Next. Eagles three. So then we can say that. See, it is equal to one plus square root of 19 over three.

Fahad Paryani · Answer

all right. So verified that the function terrified the three parties is a broad term on the given interval. And then we&#x27;re being asked to find all the numbers, sees that satisfying the conclusion of growth. So our first question is, whether is dysfunction continues because these are the three conditions required for roast room. And, um, if we look closely, we have a one over X and, um, one of her dysfunctions, actually not continuous at X equals zero. Could the few program X equals zero? You get zero plus one over zero. So you get Teo plus one over radio, and any number divided by zero does not. It goes off into positive or negative infinity, depending on where you coming in from. But however, since we are not in since our actual interval does not include zero, we don&#x27;t even have to worry about this. So this all those true did not matter. So we know that it is continuous everywhere except X equals zero. So in this case is one half to two, which means it is also continuous and one half to two and then centralised, continuous. And from one half to two, we can also assume that is also defensible in this interval and then our last conditions, we have to track its whether Africa one half tickle toe too. And if you plug in the value of one, have you get two point five and if you plug it in for athletic too, you want to get two point five? So now that all three conditions of our function has been satisfied, we can now find the we can actually not find the value of number Seas have certified his conclusion. So all we have to do now is take the derivative of primary Rex terms at that equal to Teo. And since this is a rational function which is also polynomial, you can turn this into thanks. Plus, thanks to the minus one, since that is a property of a X exponents. And so this will give us one minus X minus two and we&#x27;re being asked to set this equals zero. Will you bring accident negative two on boats to the right side. Since that is a negative and our experiment, we put it to the bottom and then we X squared equals one. And now we just take the scribble it on both side a squared of one. Get just two incher and get us positive and negative one, since our interval is all is in the positive side. So from one half to two we don&#x27;t have to worry about negative ones. So R. C that satisfies this. It&#x27;s just positive one.

Brenda Knauber · Answer

the function. Powerful Rex Eagle two. The square root over smile honest. 1/3 of home, three times X on the interval from 0 to 9 is continuous from internal containing the points that is to X equals era and defensible elementary will not continue. Then points that is, extra Klay greater than zero, so it satisfies. So let&#x27;s function almost interval satisfies the 1st 2 hypotheses of rolls through um so how many attract the third hypothesis about this? Several through, which requires that the function is equal at one point. So let&#x27;s start with the function at X equals zero, which we a spirit of zero minus 1/3 times zero, which is equal deserve and the function at X equal to nine will be the square root of nine, which is three minus one times three time, 1/3 times nine, which is also equal to three. So three minus for using called a zero. So then that satisfies all three hypotheses of roles term so that now we must be able to find too, according to their at least one function between zero. At least one aurelio of X between 09 were the function has a slope of zero. Fans were looking for a prime of C equals zero. We&#x27;re looking for that see value. So let&#x27;s take the derivative over function out of both long hair times one or roof square root of X, minus 1/3 equals zero. Move over the 1/3. Let&#x27;s put the whole fund. Let&#x27;s flip this whole function to look for X. So then two times. Skirt of X equals three derived by two and square both sides. So then X is equal to nine force, and that is between 09 cylinder. Conceal that Sequels time horse.

Fahad Paryani · Answer

Okay, So the question we&#x27;re being asked to verify that the function of satisfying three high positive roles and begin interval. And then we&#x27;re asked to find the number of sees the status by the conclusions of the rules doom. So the function here is after practical ex Cubans to X squared minus four plus two. So here we know that a Mexican ex humanity experiment for experts to is a polynomial. So we know that all polynomial is continuous and also are all defensible because those air properties upon our muse. So we know automatically now these first two conditions of roast and the satisfied and this is the last condition is where the end points are equal to each other. And if you actually plug in the values of of negative too, you get negative six. And if you plug it in for half of to, you also get negative six and you can check those values yourself and that again. Now that they&#x27;re all satisfied, we can assume that the conclusion is also food in the conclusion there is a number. See, such that that there&#x27;s a number c in the derivative of the function, so that it is. It is equal to zero. So we can actually solve for this function because we confined the derivative of affect. This is a very easy wanted to is very easy to do because it is a problem for me. So f Crime of X is equal to three x squared minus for X miners for and all we&#x27;re doing now is asked to find where dysfunction equals zero. And this is a polynomial. So, uh, you can yes, you have to factor this. And so there are numerous ways to factory was function. I do it with the expended. You may have your own methods. So all you do is multiply ity and numbers are four time negative. Three times negative forgives us. Ah, negative twelve. And then we take the middle number, which is negative for. And then we find two numbers that multiply to give us negative phone. But add to give his naked for and in this case, in his negative six and positive too. And since this is a has a leading coefficient in front of the ah of ah, first number and and this polynomial we divide that way we divide that by the leading number. And then this gives us a simplified fraction of negative dory over one. That means we take the bottom number, which is one so be one X minus three times and this function is the most simplified fraction. So you just take the bottom for number in four acts in front of three X plus two vehicle two zero. This is the factored form of there are derivative. And now all we do is fine where these individuals terms equal dear, and they equal Geo when X is equal to three. I could go in the stall for this function in history, and then he saw for three exposed to it because three which gives us well, it gives us, I believe, negative two thirds police. So those are too, uh see or X values that give us where the derivatives equal to zero. And no, this is a very, very important point about rolls term. We&#x27;re all storms only tells us that there is at least one see, so we know that there&#x27;s a least one see for sure. But in this case, we have more than once that we have to, in fact, and this is an important point because all roads there tells us that there is one that there&#x27;s at least one there could be more. Well, that could be one, two, three, four and infinitely many. And that is just an important point that should be understood of that role. Stone is that it proved that there&#x27;s on ly that there&#x27;s at least one if these conditions are satisfied.

Problem

Verify that the function satisfies the three hypo…

Problem 1 Easy Difficulty

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers $c$ that satisfy the conclusion of Rolle's Theorem.
$f(x)=5-12 x+3 x^{2}, \quad[1,3]$

Answer

$c=2$

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James Stewart

Essential Calculus Early Transcendentals

Catherine R.

Anna Marie V.

Heather Z.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

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$c=2$

Essential Calculus Early Transcendentals

Catherine R.

Anna Marie V.

Heather Z.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartEssential Calculus Early Transcendentals

Catherine R.

Anna Marie V.

Heather Z.

Joseph L.

James Stewart

Essential Calculus Early Transcendentals