Question

Find two positive numbers satisfying the given requirements. The sum is 120 and the product is a maximum.

Short answer

The two numbers are 60 and 60.

Step-by-step solution

Formulating the problem

Let the two numbers be \( x \) and \( y \). According to the given conditions, we have two equations: \( x + y = 120 \) and we need to maximize \(xy\), which is the product of the numbers.

Expressing the product in terms of one variable

From \( x + y = 120 \), we can express \( y \) in terms of \( x \) as \( y = 120 - x \). Substituting this into the product we are trying to maximize gives \( P = x(120 - x) = 120x - x^2 \).

Differentiating and finding the maximum product

Differentiate \( P \) with respect to \( x \) to get \( P' = 120 - 2x \). Set the derivative equal to zero and solve for \( x \). This yields \( x = 60 \).

Determining the values of x and y

Substitute \( x = 60 \) into the equation \( x + y = 120 \) to get \( y = 60 \).

Verifying the maximum value

Taking the second derivative of \( P = 120x - x^2 \) yields \( P'' = -2 \) which is less than zero, confirming that \( P \) is at its maximum when \( x = 60 \).