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What do we find infimum of exp(-x) without graph?
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#2
(Original post by Infinity...)
I know it is 0 but how to find it without graph?
I know it is 0 but how to find it without graph?
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(Original post by mqb2766)
You could show its a decreasing monotonic function (-ive gradient), so the minimum value must be when ...
You could show its a decreasing monotonic function (-ive gradient), so the minimum value must be when ...
to find supremum without using graph?
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#4
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...
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(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...
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...
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#6
(Original post by Infinity...)
hence 0 is lower bound. And how do we find supremum of exp(-x)?
hence 0 is lower bound. And how do we find supremum of exp(-x)?
There is no supremum of exp(-x) over R. Do you mean some other set?
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#7
(Original post by Infinity...)
yeah it wil be its limit point which is 0. May i know how
to find supremum without using graph?
yeah it wil be its limit point which is 0. May i know how
to find supremum without using graph?
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(Original post by mqb2766)
Opposite direction for x? Again use the fact its monotonic.
Opposite direction for x? Again use the fact its monotonic.
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(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?
No. Hence 0 is the infimum...
There is no supremum of exp(-x) over R. Do you mean some other set?
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#10
(Original post by Infinity...)
So is it not bounded above because it is monotonic?
So is it not bounded above because it is monotonic?
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#11
(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?
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?
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#12
(Original post by Infinity...)
So is it not bounded above because it is monotonic?
So is it not bounded above because it is monotonic?
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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