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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 - 75 $; Maximum :25

$b=\pm 20$

Algebra

Chapter 2

Polynomial and Rational Functions

Section 1

Quadratic Functions and Models

Quadratic Functions

Complex Numbers

Polynomials

Rational Functions

Missouri State University

McMaster University

University of Michigan - Ann Arbor

Lectures

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let'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's go ahead and do that. So let me take the original problem that we have. All right? For this first step, I'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't done anything yet. Just rewrote this. So now I'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's what we'LL add in here. And however, if we're going to add something we have to make up for by cancelling it out So notice here that we didn't really add b squared over for because of this minus, we actually subtracted it. So we'll make up for this attraction by just adding it back in. So now that we'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'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'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'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's our final answer.

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