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Chapter 2: Problem 71

Find two positive real numbers whose product is a maximum. The sum is 110 .

Short Answer

Expert verified
The two positive real numbers whose sum is 110 and whose product is a maximum are 55 and 55.

Step by step solution

01

Formulate the problem

Let's use \( x \) and \( y \) as our two positive real numbers. Then the given conditions translate to the following:1. \( x + y = 110 \) (condition for their sum)2. \( xy = P \) (we'll call the product \( P \))
02

Express y in terms of x

From the first condition, we can find \( y = 110 - x \). This will allow us to express the product only in terms of \( x \).
03

Formulate the product function

Rewrite the product \( P \) using the equation for \( y \) from the previous step: \( P = x(110 - x) = 110x - x^2 \). This is a quadratic function, and it will have a maximum value.
04

Find the maximum product

To find the maximum of the function, take the derivative and set it to 0. The derivative of \( P = 110x - x^2 \) is \( P' = 110 - 2x \). Setting this to 0 gives \( x = 55 \). Substituting this back into \( y = 110 - x \) gives \( y = 55 \) too. So the maximum product occurs when both numbers are 55.

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