Prove that it is impossible to maximize the sum of two real numbers if their product is 4.

Answer 1

xy = 4

y=4/x

S = sum = x+y = x + 4/x

take the derivative and set equal to zero then solve for x

S' = 1 -4/x^2 = 2

4/x^2 = 1

x^2 = 4

x= +2 or -2 and y = +2 or -2 those are the values that give the minimum sum

Graph xy =4 and you get a rectangular hyperbola, half in the 1st quadrant, half in the 3rd quadrant

minimum sum is at the points (2,2) or (-2,-2)

maximum sum is at the "endpoints" of the two branches, but there are no "endpoints. the x and y axes are asymptotes and the graph never touches them.

Maximum "points" are (0, infinity), (0, -infinity), (infinity, 0) and (-infinity, 0) but those aren't real points. They're "surreal" points. Just limits that are never reached. So there is no maximum

(2,2) gives a sum of 4

(4,1) gives a sum of 5

(8, 1/2) gives a sum of 8.5

(16, 1/4) gives a sum of 16.25

....

(4 million, 1/1 million) gives a sum of slightly over 4 million

....

(infinity, 1/infinity) gives a sum of infinity, the "maximum" sum of 2 numbers whose product can be 4