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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

$b=\pm 16$

Algebra

Chapter 2

Polynomial and Rational Functions

Section 1

Quadratic Functions and Models

Quadratic Functions

Complex Numbers

Polynomials

Rational Functions

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McMaster University

Harvey Mudd College

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let'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'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'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'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'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's, we could complete that square and then we're just left over with our constant term. Here'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'LL give you sixty four. So then go ahead and multiply that foreign there. So we'LL have hoops B square equal. So here let'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's two different values of B using either one of these values. If you go to the formula for the Vertex, you'LL see that in each case, the K value is for forty eight, so there's our final answer.

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