Let p + q = 10 , where p , q are integers. Value-I = Maximum value of p × q when p , q are positive integers. Value-II = Maximum value of p × q when p ≥ - 6 , q ≥ - 4 . Which one of the following is correct?

To find the maximum value of p * q when p + q = 10, we can express q as 10 - p. The expression becomes p * (10 - p) = 10p - p^2. This is a downward-opening parabola, and its maximum value occurs at the vertex, which is p = -10 / (2 * -1) = 5. When p = 5, q = 10 - 5 = 5. The product p * q is 5 * 5 = 25. Value-I: Maximum value of p * q when p, q are positive integers. The integers p and q must be positive and sum to 10. To maximize their product, they should be as close as possible. The closest positive integers are p = 5 and q = 5. For p = 5, q = 5: p * q = 5 * 5 = 25. All other pairs of positive integers (e.g., 4, 6 or 3, 7) will yield a smaller product. So, Value-I = 25. Value-II: Maximum value of p * q when p >= -6, q >= -4. We already established that the absolute maximum of p * q for p + q = 10 occurs at p = 5, q = 5, giving a product of 25. We need to check if these values satisfy the given constraints: For p = 5: Is p >= -6? Yes, 5 >= -6. For q = 5: Is q >= -4? Yes, 5 >= -4. Since p=5 and q=5 satisfy the constraints and yield the absolute maximum product, Value-II is also 25. Comparing the values: Value-I = 25 Value-II = 25 Therefore, Value-I = Value-II. The correct option is C.