Question

among all pairs of numbers whose difference is 16, find a pair whose product is as small as possible. what is the minimum product? the pair of numbers whose difference is 16 and whose product is as small as possible is. (use a comma to separate answers.)

Explanation:

Step1: Define Variables

Let the two numbers be \( x \) and \( y \), with \( x - y = 16 \), so \( x = y + 16 \). The product \( P = x \cdot y=(y + 16)y=y^{2}+16y \).

Step2: Find Minimum of Quadratic

The quadratic function \( P(y)=y^{2}+16y \) has \( a = 1 \), \( b = 16 \), \( c = 0 \). The vertex of \( ay^{2}+by + c \) is at \( y=-\frac{b}{2a} \). So \( y =-\frac{16}{2\times1}=- 8 \).

Step3: Find Corresponding \( x \)

Substitute \( y=-8 \) into \( x = y + 16 \), we get \( x=-8 + 16 = 8 \).

Answer:

\( 8, - 8 \)