Question

Let X =Y = { x in R: 0 < x<1 } Define h: XxY -> R by h(x,y) = 2x + y

a. for each x in X find f(x) = sup{h(x,y): y in Y} then find inf {f(x): x in X}
b. for each y find g(y) = inf {h(x,y): x in X} then find sup {g(y): y in Y}.

Answer 1

Ok, so, is f(x) clear? Can you write it down?

Answer 2

f(x) = sup{ 2x + y | y in (0,1) } hence f(x) = ....

Answer 3

yes that is f(x) it is thwe result when you map x,y onto h(x)

Answer 4

I want you to see there is way to write a formula for f(x) that doesn't involve sup or inf.

Once you have formula, we can then deal with the second part of inf{f(x) : ....}

Answer 5

If you're not seeing this, let me ask you this. What is

sup{ y | y in (0,1) } ?

Answer 6

yes that is f(x) it is thwe result when you map x,y onto h(x)

Answer 7

f(x) = sup (2x +y) = 2x + 1 since y 0<y<1

Answer 8

Right. Hence

inf{ f(x) | x in (0,1) } is what?

Answer 9

yes that is f(x) it is thwe result when you map x,y onto h(x)

Answer 10

0

Answer 11

Nope

Answer 12

inf{ f(x) | x in (0,1) }
= inf{ 2x + 1 | x in (0,1) }
= ...

Answer 13

yes that is f(x) it is thwe result when you map x,y onto h(x)

Answer 14

oops of f(x) = inf (2x +1 ) = 1

Answer 15

right.

I'll you work through the second part.

Answer 16

I'll let you work through the second part.

Answer 17

yes that is f(x) it is thwe result when you map x,y onto h(x)

Answer 18

g(y) = y

Answer 19

yes that is f(x) it is thwe result when you map x,y onto h(x)

Answer 20

sup(g(y)) = 1

Answer 21

yes

Answer 22

yes that is f(x) it is thwe result when you map x,y onto h(x)

Answer 23

coolio and beans. thanks. now I have to do it in another problem for h (x,y) = 0 if x << y or 1 if x[\le\] y. just vary the x for f(x) or y for g(y) and just follow through as above?

Answer 24

Put this in a new post and make sure you have a well posed/defined question, as I'm not following exactly what you're saying.

Answer 25

yes that is f(x) it is thwe result when you map x,y onto h(x)

Answer 26

regardless of what function h(x,y) is, would the reasoning to find inf (f(x)) and sup g(y) the same?

Answer 27

In general, yes.

Answer 28

yes that is f(x) it is thwe result when you map x,y onto h(x)

Answer 29

thanks again for the walk through. if you can throw any hints at the other posted question that would be cool.