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**1) Find the global max and min values of the function**

f(x,y)=x/[x

f(x,y)=x/[x

^{2}+(y-1)^{2}+4] on the first quadrant S={(x,y)|x,y__>__0}__Solution: (from textbook example)__

f(x,y)

__>__0 on S and f(0,y)=0, so the minimum is zero.

Moreover, f(x,y) is less than the smaller of 1/x and 1/(y-1)

^{2}, so f vanishes as |(x,y)|->∞. Hence, by theorem (*), f has a maximum on S. This justifies the "existence" of an absolute max on S.

...

theorem (*): Let f be a continuous function on an unbounded closed set S in R

^{n}. If f(x)->0 as |x|->∞ (x E S) and there is a point xo E S where f(xo)>0, then f has an absolute maximum on S.

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I understand why f(x,y) is less than 1/x, but I can't figure out why f(x,y) is less than 1/(y-1)

^{2}. Can someone help me, please?

Secondly, why does f vanish as |(x,y)|->∞? There are many different paths for the

**length**to go to infinity, e.g. x can go to infinity while y is 0 (along x-axis), or y can go to infinity while x is 0 (along y-axis), or x and y may both go to infinity, how can we be sure that in

**ALL**directions to infinity, f will vanish?

Thanks anyone for explaining!

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