Hello there, I have this question that comes with a hint; however the hint doesnt seem to make any sense to me whatsoever so I would appreciate some insight into this.
Let f
be a function. We define
0 as
to mean that
such that
Similarly define
as
to mean that
such that
Now suppose that
is continuous and has
as
and
as
. Show that f is bounded and attains its maximal value on
, i.e. there is some
with
.
[HINT: If f(x) = 0 for all x, there is nothing to prove (why not?
[well because f(x)= a constant so it is bounded by that and that is its maximal value for all x] so we can assume that there is some with f(x) > 0. Then take and find such that and for or . Now use the extreme value theorem ] Solution: or as much as I can do...
Now
such that
.
Let
. Choose
such that
.
Choose
such that
.
But
which is a contradiction.
And
which is a contradiction.
Hence
.
But this doesn't seem to show me anything?? Any help with this would be deeply appreciated.