of all numbers whose difference is 18, find the two that have the minimum product. what are the two numbers…

of all numbers whose difference is 18, find the two that have the minimum product. what are the two numbers? (use a comma to separate answers as needed.)

of all numbers whose difference is 18, find the two that have the minimum product. what are the two numbers? (use a comma to separate answers as needed.)

Answer

Explanation:

Step1: Set up variables

Let the two numbers be (x) and (y). Assume (x>y), so (x - y=18), then (x=y + 18). The product (P=xy=y(y + 18)=y^{2}+18y).

Step2: Find the minimum of the quadratic function

For a quadratic function (P(y)=ay^{2}+by + c) ((a = 1), (b = 18), (c = 0)), the vertex of the parabola (y=-\frac{b}{2a}). Substitute (a = 1) and (b = 18) into the formula (y=-\frac{18}{2\times1}=-9).

Step3: Find the other number

Since (x=y + 18), when (y=-9), (x=-9 + 18=9).

Answer:

(9,-9)