Among all pairs of numbers whose sum is 4 , find a pair whose product is as large as possible. What is the maximum product

1. Let the pair of numbers be \( x \) and \( y \).
2. We have the constraint \( x + y = 4 \).
3. The product \( P \) is given by \( P = xy \).
4. Substitute \( y = 4 - x \) into the product equation:
P = x(4 - x)
5. Simplify to get the quadratic function:
P = 4x - x^2
6. To find the maximum product, take the derivative \( \frac{dP}{dx} \) and set it to zero:
\frac{dP}{dx} = 4 - 2x
4 - 2x = 0
x = 2
7. Substitute \( x = 2 \) back into the sum equation to find \( y \):
y = 4 - 2 = 2
8. The pair that gives the maximum product is \( (2, 2) \).
9. The maximum product is:
P = 2 \times 2 = 4

Therefore, the maximum product is \text{Maximum Product} = 4 .