Questions: What two nonnegative real numbers with a sum of 32 have the largest possible product? The numbers that have a sum of 32 and have the largest possible product are.

Solution Steps

Let \( x \) and \( y \) be the two nonnegative real numbers such that \( x + y = 32 \).

The product \( P \) of the two numbers can be expressed as: \[ P = x \cdot y \]

Using the equation from Step 1, we can express \( y \) in terms of \( x \): \[ y = 32 - x \] Substituting this into the product equation gives: \[ P = x \cdot (32 - x) = 32x - x^2 \]

To find the maximum product, we can take the derivative of \( P \) with respect to \( x \) and set it to zero: \[ \frac{dP}{dx} = 32 - 2x \] Setting the derivative equal to zero: \[ 32 - 2x = 0 \]

Solving the equation from Step 4: \[ 2x = 32 \implies x = 16 \]

Using the value of \( x \) to find \( y \): \[ y = 32 - x = 32 - 16 = 16 \]

The two nonnegative real numbers that have a sum of 32 and yield the largest possible product are \( x = 16 \) and \( y = 16 \).

Final Answer