I want to know the maximum and minimum value of this function x/(x+y). I tried the usual methods but i was not able to get it. So please suggest me some alternative method that will be help to find maxima and minima of this function.
It should be specified on which set of pairs (x,y) you work. For example, if (x,y) is in the first quadrant, and x is not zero, we have x / (x+y)0. Making x to converge to zero, or y to infinity and x bounded, we see that the minim is 0, which is not attained. at any point. If you consider the domain as being the whole plane, then the maximum is +infinity, and the minim is - infinity (work around the subset (x, - x)).
If x and y are both positive integers, then f has positive (rational) values. Keeping x fixed and increasing indefinitely y through infinity, one deduces inf (f) =0, value which is not attained. So, there are no " minimum points ". On the other hand, x / (x+y) < x/x =1= max (f) for all positive integers x,y, so that all points (x,x) (where x is a positive integer) are " maximum points " for f.
Dear Colleague,
There is an error in my preceding answer : at a point (x,x) the value of f is 1/2 < 1 = sup f . So, if we work with positive integers, then sup f =1 is not attained at any point (x,y).
Sincerely, Octav
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