4. (5pts) find two positive numbers, x and y, whose sum is s = x + y = 24 and their product p = xy is…

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 ).