of all numbers whose difference is 16, find the two that have the minimum product.\nwhat are the two…

Explanation:

Step1: Define variables

Let the two numbers be (x) and (y). Given (x - y=16), so (y=x - 16). The product (P=xy=x(x - 16)=x^{2}-16x).

Step2: Find the vertex of the quadratic function

For a quadratic function (y = ax^{2}+bx + c) ((a = 1), (b=-16), (c = 0) in (P=x^{2}-16x)), the (x) - coordinate of the vertex is given by (x=-\frac{b}{2a}). Substitute (a = 1) and (b=-16) into (x=-\frac{b}{2a}), we get (x=-\frac{-16}{2\times1}=8).

Step3: Find the value of (y)

Since (y=x - 16), when (x = 8), (y=8-16=-8).

Answer:

(8,-8)