Problem 71 Find two positive real numbers w... [FREE SOLUTION]

Chapter 2: Problem 71

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

Short Answer

Expert verified

Both numbers are 55.

Step by step solution

Identification of Variables

Let's denote the two positive numbers as \(x\) and \(y\). The problem states that their sum is 110 . So, we have: \(x + y = 110\)

Setting Up the Equation

Since we know the sum of the two numbers is 110, we can express one variable in terms of the other: \(y = 110 - x\). The product of two numbers is \(P = x * y = x * (110 - x)\), which is the function we want to maximize.

Taking Derivative

The next step is to find the derivative of the function \(P\). We will use the power rule, which states that the derivative of \(x^n\) is \(n*x^{n-1}\). The derivative of \(P\) with respect to \(x\) is: \(P' = 110 - 2x\).

Finding the Critical Numbers

Now, we find the critical points where the derivative is zero or doesn't exist. \(110 - 2x = 0\). Solving for \(x\), we get \(x = 55\). As \(x + y = 110\), substituting \(x = 55\) into the equation, we obtain \(y = 55\). So the numbers are 55 and 55.

Confirming the Maximum Value

To confirm that the product is a maximum, we need to verify the second derivative. The second derivative of \(P\) is: \(P'' = -2\). Since -2 < 0, this indicates a maximum. Thus, the product of the numbers is indeed maximum when the numbers are both 55.

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91影视

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

Short Answer

Step by step solution

Identification of Variables

Setting Up the Equation

Taking Derivative

Finding the Critical Numbers

Confirming the Maximum Value

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