Questions: Among all pairs of numbers whose sum is 14 , find a pair whose product is as large as possible. What is the maximum product?

Solution Steps

To find the pair of numbers whose sum is 14 and whose product is as large as possible, we can use the concept of optimization. The problem can be formulated as maximizing the product \( x \times y \) subject to the constraint \( x + y = 14 \). By substituting \( y = 14 - x \) into the product expression, we can derive a quadratic function in terms of \( x \) and find its maximum value using calculus or by completing the square.

We need to find two numbers \( x \) and \( y \) such that their sum is \( 14 \) and their product \( P = x \cdot y \) is maximized. This can be expressed mathematically as: \[ x + y = 14 \] \[ P = x \cdot y \]

Using the constraint \( y = 14 - x \), we can rewrite the product as: \[ P = x \cdot (14 - x) = 14x - x^2 \]

To maximize the product, we take the derivative of \( P \) with respect to \( x \): \[ \frac{dP}{dx} = 14 - 2x \] Setting the derivative equal to zero to find critical points: \[ 14 - 2x = 0 \implies x = 7 \]

Substituting \( x = 7 \) back into the equation for \( y \): \[ y = 14 - x = 14 - 7 = 7 \]

Now, we can calculate the maximum product: \[ P = 7 \cdot 7 = 49 \]

Final Answer