in-exercises-91-94-find-the-values-of-such-that-the-function-has-the-given-maximum-or-minimum-valu-2

Question

In Exercises 91- 94, find the values of such that the function has the given maximum or minimum value.

$ f(x) = -x^2 + bx - 16 $; Maximum: 48

J Hardin · Accepted Answer

let&#x27;s find the values of B where the maximum of this quadratic is forty eight. So we know that this is a quadratic that opens downward due to the negative leading coefficient. So what we really want here is we have our Vertex. Let&#x27;s say each calm okay, and the y value there, that will be the highest value on the graph. So we want that to be forty eight. So what we should do here is look at the standard form. And the reason for that is because if you write up in the standard form, then this constant term hanging on side by itself, that is the K that is the max value for the for this problem. So in our case, let&#x27;s just take our original function here and just do some math here. So let me just go ahead and pull out a negative from the first two terms and just go ahead and leave that minus sixteen outside and then here will go ahead and complete the square inside the parentheses. So we take our negative, be divided by two square it, and that gives you be squared over for So this is what we should be adding it. So let&#x27;s be careful now. We can just go ahead and and add this in without making up for it. And also a note we didn&#x27;t really added it. We really subtracted and due to the minus sign outside. So if we wantto balance this out, we need to go ahead and add it back in so that we have not changed the problem. These three terms one with the negative one Positive. Those two would cancel out that the reason for doing this is that now, inside the Prentice&#x27;s, we could complete that square and then we&#x27;re just left over with our constant term. Here&#x27;s r k hanging outside by itself. In our case, it be squared over four minus sixteen. So this is our max value. So this is what we should be setting equal to forty eight. So our new equation tau look at is p squared over four minus sixty nickels for years. And solving for this gives us our B values so we can go ahead and at that sixteen over that&#x27;LL give you sixty four. So then go ahead and multiply that foreign there. So we&#x27;LL have hoops B square equal. So here let&#x27;s just do four times sixty four and then when we go to be so here, we should do plus or minus and then scream for square sixty four, giving you plus or minus. And that here will have to. And then this will be eight so we can see here by using the values closer minus sixteen, so these will correspond to two different. These are two different travel. It&#x27;s two different values of B using either one of these values. If you go to the formula for the Vertex, you&#x27;LL see that in each case, the K value is for forty eight, so there&#x27;s our final answer.

J Hardin · Answer

So first, let&#x27;s he&#x27;s the fact that the Vertex is given by the point negative B over two a and then f of negative B over two a were in general your problems of this form. So we want to find the value of B that makes the minimum value. So that will be the white coordinator of the first hex. So just a quick picture was going on here. Our leading coefficient is positive, so we know the problem opens upward. And if you go to the Vertex, the Y coordinate of that is the minimum value. And that&#x27;s all we&#x27;re looking for here. And we want that to be minus fifty. So we want this value over here to be minus fifty. So first, let&#x27;s find negative B over two A. So in general bees, the coefficient in front X here it&#x27;s also happens to be B and then our is equal to one. So we have negative B over two times one so just negative B over two. And so now we plug this into F or if it&#x27;s just our original quadratic. So first you square that I will give you be squared over for plus B times X minus twenty five and we want this toe all equal the minimum which is minus fifty. So we have b squared over four minus B, squared over two minus. And then let me go ahead and take that negative twenty five and then add it to the other side. So let me take a few steps back here. Sorry about that. So the left hand side, you can go ahead and get a common denominator on the left that would be be squared over four minus to be split over for which is negative. B squared over four. So the final equation we had over here becomes minus B squared over four equals minus twenty five. Good. And multiply both sides by negative for. And when you do that, you have two values from B that work. So by using the value closer minus ten, either one of these values and plugging it in for this be appear that will ensure that the minimum value is minus fifty

J Hardin · Answer

let&#x27;s find the value or values of B that such that the maximum of this quadratic is twenty five. So if we look at this coefficient, the leading coefficient is negative one. So this tells us that the problem would open downward. And so what this means is that the Max point, the largest y value, will just be the second coordinate here himself. The Vertex is h k. We want Kate would be twenty five. So the way to do this is will rewrite our quadratic in the standard form or H in case the Vertex. And we want to set this up in such a way such that the case equal to twenty five. So let&#x27;s go ahead and do that. So let me take the original problem that we have. All right? For this first step, I&#x27;m just going to factor out a negative one and let me on. Lee, just do this for the first two terms, so have a X squared minus B x and then my two seventy five outside. So we haven&#x27;t done anything yet. Just rewrote this. So now I&#x27;ll complete the square, so we have to go ahead and add something inside this Prentiss see? So you take the term in front of the ex, which is negative b you divided by two and then you square that to get b squared over for So that&#x27;s what we&#x27;LL add in here. And however, if we&#x27;re going to add something we have to make up for by cancelling it out So notice here that we didn&#x27;t really add b squared over for because of this minus, we actually subtracted it. So we&#x27;ll make up for this attraction by just adding it back in. So now that we&#x27;ve completed the square and the Prentice is, we can go ahead and write this Just go ahead and factor that quadratic in the parentheses and then the remaining term that we have we&#x27;LL have B square over four minus seventy five Let me go ahead and just get a common denominator. You know what? Let me just keep this how it is. Sorry. So we want If you look at this expression over here, this&#x27;s RK and we want came to be equal twenty five. So we should be looking at B Square over four minus seventy five. Since this is our K and we want that Katie to be twenty five. There we go. And then just go ahead and solve this for B. So add the seventy five over, multiply both sides by four, and then take the square root of both sides. And don&#x27;t forget the negative route as well. You have plus or minus twenty. So taking B to B twenty year minus funny. These were the two values that will ensure that their maximum of this quadratic is twenty five. And that&#x27;s our final answer.

J Hardin · Answer

here is our parabola and you see that the leading coefficient is positive. It&#x27;s one. So this means that our problem will have been upward and that our birthday cakes the point h calm Okay, since it opens up, K will be the minimum value. And in our case, we want this to be ten. So we can do here is rewrite our function in the standard form. So what I&#x27;ll do is we take the term in front of X, which is B divide that by two in the army and then square it. However, to make up for this, we also have to subtract the same expression. So we&#x27;LL have X squared the ex b squared over four twenty six minus B squared over for and then and the apprentices we complete the square Now we are in the Vertex form The standard form recall that&#x27;s equal to this is the standard form and in this case, the hur Texas given by the point h com McKay. So in our case, we see that r K value out here on the side agent is given by this expression and the problem. They&#x27;re telling us that this is equal to ten. So we&#x27;re trying to find the value of B that makes this expression on the left equal to ten. So let&#x27;s just go ahead and solve this for B. So we have peace. Where is sixty four? And here we have two values of B plus or minus square room sixty four. And so the either of these values of B will ensure that the minimum value is ten.

Ankit Gupta · Answer

on anyone. This was any other find other than your visas that Montana four packs on maximum their new 40. So we know maximum will get it. Go to a minus bi bi to get so mine It&#x27;s me But even if you take that minus off the videos be only you are very were made to go into a release minus one like this in providing this indicative that could be a breakthrough So I will get exported The bagel Thanks a lot. So if I reported yet excellent be by group in there for if I do this only the maximum before the a minus the vital all this way Let&#x27;s be in till we invite toe find at 16 about This is the one who is 14 years off the 16. If I decided that plus 16 report of the person who read an elegant piece where there are four knowledge he had this one then under six before although these were vying for for that. If I send the bird and Beast Medical Ground 64 into 495 minutes later on both sides and people will get let&#x27;s minus 16. So this is not really all be for this undeclared decent, so dishonest

Amrita Bhasin · Answer

this question strikes us to find them a moment back soon. Values of the function subject to the given constraint. What we know is that we have the LaGrange equations. Why is eight Lambda Axe and then X is 18? Wendell? Why? Which means that Lambda is why Divided by eight packs. Why divide by eight x, which is the same thing as acts divided by 18. Boy, not solving these above. Equations. We get four. We get. Why, over a axes X over 18. Why? Which means now we can solve the equations for X and y using the constraint, we have axes. Poster minus two. Okay, not F of two comma. 4/3. Plug it in to calculate the critical values. And you get 8/3 off of negative two comma, 4/3 plugging in gives us negative eight divided by three. This is our maximum. And this is our minimum

Problem

In Exercises 91- 94, find the values of such that…

Problem 92 Hard Difficulty

In Exercises 91- 94, find the values of such that the function has the given maximum or minimum value.

$ f(x) = -x^2 + bx - 16 $; Maximum: 48

Answer

$b=\pm 16$

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$b=\pm 16$

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