What do we find infimum of exp(-x) without graph?
I know it is 0 but how to find it without graph?
Original post by Infinity...
I know it is 0 but how to find it without graph?
You could show its a decreasing monotonic function (-ive gradient), so the minimum value must be when ...
Original post by mqb2766
You could show its a decreasing monotonic function (-ive gradient), so the minimum value must be when ...
yeah it wil be its limit point which is 0. May i know how
to find supremum without using graph?
Original post by Zacken
Just use the definition, you know exp(-x) > 0. If e > 0 is a lower bound then take x = ln(2/e) so that exp(-x) = e/2. So e > 0 can't be a lower bound. Hence...
hence 0 is lower bound. And how do we find supremum of exp(-x)?
Original post by Infinity...
hence 0 is lower bound. And how do we find supremum of exp(-x)?
No. Hence 0 is the infimum...
There is no supremum of exp(-x) over R. Do you mean some other set?
Original post by Infinity...
yeah it wil be its limit point which is 0. May i know how
to find supremum without using graph?
to find supremum without using graph?
Opposite direction for x? Again use the fact its monotonic.
Original post by mqb2766
Opposite direction for x? Again use the fact its monotonic.
So is it not bounded above because it is monotonic?
Original post by Zacken
No. Hence 0 is the infimum...
There is no supremum of exp(-x) over R. Do you mean some other set?
There is no supremum of exp(-x) over R. Do you mean some other set?
Yes, there.is no upper bound of exp(-x). Is there some way to prove this or it is just the nature of this function?
Original post by Infinity...
So is it not bounded above because it is monotonic?
No, it's not bounded above because it's not bounded above. Not because it's monotonic.
Original post by Infinity...
Yes, there.is no upper bound of exp(-x). Is there some way to prove this or it is just the nature of this function?
Why can't you just go back to the definition? If A > 0 is an upper bound of exp(-x) then just pick x = - ln(2A) so that exp(-x) = 2A > A. So there is no upper bound for exp(-x).
Original post by Infinity...
So is it not bounded above because it is monotonic?
Yes (assuming x -> -inf).
For a bit more thorough analysis you need to show the gradient doesn't go to zero as well (which happens when x-> inf) as otherwise it would converge to some value
But if you have a monotonic function, the min max values must occur at the ends?
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