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