Find two positive numbers whose product is 192 and sum is a minimum.

Question

How do you solve this without using derivatives and using Pre-calc

Answer 1

The unknowns are X and Y. We are told that X*Y = 192......so Y = 192/X

The sum is therefore S = X + 192/X

WE can then find the minimum by graphing. For example, try the DESMOS graphing tool.....this shows a discontinuity when x --> 0, and minimum absolute values for S when X = + or - 14

Answer 2

We can use the arithmetic inequality: For any non-negative numbers a and b, a+b>=2sqrt(ab) with the equality being held if and only if a=b.

Let x and y be two positive numbers with xy=192. Our goal is to minimize the value x+y, that is x+(192/x).

Now by the arithmetic inequality x+(192/x)>=2sqrt(x*(192/x))=2sqrt(192)=16sqrt(3). This means, the least value of x+(192/x) is 16sqrt(3); this minimum is obtained when x=(192/x) or x^2=192 which yields x=sqrt(192)=8sqrt(3).

Now we conclude that the two numbers are 8sqrt(3) and 8sqrt(3), and the minimum value of their sum is 16sqrt(3).