find positive numbers x,y with xy such that xy=7 and x+y is as small as possible

Question

find positive numbers x,y with x>y such that xy=7 and x+y is as small as possible

blacksteel · Answer

The smallest possible value for x+y given xy=7 would be at x=sqrt(7), y=sqrt(7). This is because the value of two numbers whose product is given is minimized when they are equal, and vice versa (the product of two numbers whose sum is given is maximized when they are equal). This can be proven using the quadratic formula, which I can do if you want.

However, we have the added constraint that x>y. Then the value of x+y would be minimized at x=sqrt(7)+a, y=7/[sqrt(7)+a] for an infinitesimally small a - technically there is no direct solution for this problem.

anonymous · Answer

Or you can y=7/x  so Sum S = x+ 7/x and differentiate (x>0) -> 1-7/x^2 = 0 -> sqrt 7