We are given x + y = 10, so y = 10 - x. The product function is P(x) = x(10 - x) = 10x - x². To maximize P, we find its derivative: P'(x) = 10 - 2x. Setting P'(x) = 0 gives x = 5. The second derivative P''(x) = -2 is negative, confirming that x = 5 yields a maximum. When x = 5, y = 10 - 5 = 5. The maximum product is P = 5 * 5 = 25. Option A (20) and B (24) are products of other pairs (like 28 or 46) which are less than 25. Option D (100) is the square of the sum, not the maximum product under the constraint.