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In Exercises 71-74, find two positive real numbers whose product is a maximum.
The sum of the first and twice the second is $ 24 $.
$6 \times 12$
Polynomial and Rational Functions
Quadratic Functions and Models
Oregon State University
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for this problem will find through positive real numbers. So let's go ahead and call this X and y. We want to know Numbers X and Y that satisfied the following conditions. One is that the sum of the first and twice the second is twenty four. So let's go ahead and just call the first one x and call the second one y So the sum of X plus twice the second one. That would be, too. Why equals twenty four that's given information and the quantity that we'd like to maximizes the product. So let's go ahead and call the product. P is X times Why? So we want to maximize key. So what we can do here is just go ahead, look at our linear function here and solve this for either extra. Why, it doesn't matter which one. So if I go ahead and you solve this for X twenty four minus two y So let's go ahead and plug in this X value into this equation over here, and that gives you twenty four minus two. Why, times why? And then go ahead and distribute the Y through the parentheses and we see that we have a quadratic. So it's right that out. And we know how to find the maximum value of a quadratic that opens down work. This one opens downward because the leading coefficient minus two, is negative. So a equals minus two negative. Oops. Sorry about that. This is a negative quantity. That means that it opens downward. So the crab will look something like this, and we can go ahead and find that maximum value. So the maximum value happens when by looking at the first cordon of the Vertex and negative B over two A. So in our problem, B is twenty four. That's the coefficient infront of the Wai sum minus twenty four over two times are leading coefficient. A much is minus two. So we have twenty four over four, which is six. So this tells us that our second number, which we denoted by why has to be given by six. And now we want to go ahead and find the X value which we can use this equation for. We've already saw for X so bad and plug that in X equals twenty four minus two times why we just found why it's six. So That gives us twenty for minus twelve equal. Swell. So our final answer will be first number. This twelve second number is six. So these are the numbers that when you add the first times two times a second, you have twenty four and using the birth text formula for a quadratic. We know that this happens when Wyatt six in excess twelve. So that's our final answer.
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