Simply observe that f attains its maximum (that we know which is in the compact interval
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
Similarly define as to mean 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.
But this doesn't seem to show me anything?? Any help with this would be deeply appreciated.
Well I think the hint is a huge one, and in fact it is practically the solution: in general and from what they give you, you have that that , so the function is bounded in , and we're left only to worry about boundness of the function in ...but this is a closed bounded interval, and f is continuous, so...