Question

1. (5pts) find two positive numbers, x and y, whose sum is s = x + y = 24 and their product p = xy is maximized.

Explanation:

Step1: Express y in terms of x

Given \( S = x + y = 24 \), we can solve for \( y \):
\( y = 24 - x \)

Step2: Define the product function

The product \( P = xy \). Substitute \( y = 24 - x \) into the product:
\( P(x) = x(24 - x) = 24x - x^2 \)

Step3: Find the vertex of the quadratic function

The function \( P(x) = -x^2 + 24x \) is a quadratic with \( a = -1 \), \( b = 24 \), \( c = 0 \). For a quadratic \( ax^2 + bx + c \), the x-coordinate of the vertex (maximum, since \( a < 0 \)) is \( x = -\frac{b}{2a} \):
\( x = -\frac{24}{2(-1)} = \frac{24}{2} = 12 \)

Step4: Find y when x = 12

Substitute \( x = 12 \) into \( y = 24 - x \):
\( y = 24 - 12 = 12 \)

Answer:

The two positive numbers are \( x = 12 \) and \( y = 12 \), and the maximum product is \( P = 12 \times 12 = 144 \).