If f(c)>g(c) for some c and f(x)>g(x) implies f'(x)>g'(x) for any x then is f(t)>g(t) for all t?
#is this true, if yes how do I prove it?
sand cedar
pulsar haloBOT
If f(c)>g(c) for some c and f(x)>g(x) implies f'(x)>g'(x) for any x then is f(t)...
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sand cedar
My intuition says it has to be true
daring hemlockBOT
bathroom mug
sand cedar
The problem I have with this is that the interval (-delta,delta) could be shrinking so fast that the "induction" will only give us the validity of the statement in a finite interval instead of over the entirety of the real numbers
timber needle
If f(c)-g(c)>0 for some c, then there is an open neighborhood around c where this inequality holds
junior lodge
Does this only hold for t>c
timber needle
the second condition is funny. The converse is true for open intervals
junior lodge
I’m just thinking if f(x) = x and g(x) =0 and we pick a c say 3
F(c) > g(c) and f’(t) > g’(t) for all t but f(-5) < g(-5)
timber needle
yeah I thought about the negation too, didn't see anything wrong with it but couldn't come up with an example lol
sand cedar
I’m just thinking if f(x) = x and g(x) =0 and we pick a c say 3
F(c) > g(c) and ...
Yeah, so what's the verdict— t>c only? Also is my proof valid at all?
junior lodge
unsure youd need someone better than me to verify
wispy saffron
Any functions f and g such that f'(x) > g'(x) for all x would automatically satisfy the second condition, because f(x) > g(x) being true would always imply f'(x) > g'(x). So simple linear functions like f(x) = 2x and g(x) = x would satisfy the conditions. But f(x) is not greater than g(x) everywhere.
sand cedar
Any functions f and g such that f'(x) > g'(x) for all x would automatically sati...
Yeah, so t>c instead of everywhere
wispy saffron
Oh yeah, for sure. The integral of f' will be >= the integral of g', so f would have to be greater than g for all t>c.
sand cedar
Or rather t>inf(such c)
To get a good bound
I don't know why but I was desperately trying to make it work for all t, even though I knew it was kinda wrong. Thanks.
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faint drumBOT
Unable to parse the channel name
sand cedar
+close