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9aac582ad5fb4c04bb45b087ffa94d43
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s_chanch
01/16/2022 04:58:00
@sreedev there?
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sreedev
01/16/2022 04:58:24
Yes.
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sreedev
01/16/2022 04:58:46
Could you remind me the definition of inflexion points
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s_chanch
01/16/2022 04:58:59
f”(x) = 0
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s_chanch
01/16/2022 04:59:24
chabge of concavity
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sreedev
01/16/2022 05:01:51
we can try an intutive proof first.
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s_chanch
01/16/2022 05:02:24
if f”(x) = 0, its tangent line is both above and below the graph
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s_chanch
01/16/2022 05:02:32
and if f’(x) = 0, f is horizontal
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s_chanch
01/16/2022 05:02:42
(assume f isn’t constant)
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s_chanch
01/16/2022 05:03:15
that seems implausible
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sreedev
01/16/2022 05:03:23
Yes,true.
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s_chanch
01/16/2022 05:03:25
the only case it’s NEITHER above nor below is when f is constant
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sreedev
01/16/2022 05:03:37
Right.
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s_chanch
01/16/2022 05:04:04
wait
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s_chanch
01/16/2022 05:04:14
does having analysis knowledge make these a lot more easier lol
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s_chanch
01/16/2022 05:04:53
just asking
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sreedev
01/16/2022 05:05:01
You are saying this since f"(x)=0 implies f" changed sign before and after going throgh x(by IVT) and thus f' changed from increasing state to decreasing or vice versa right?
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sreedev
01/16/2022 05:05:54
It helps to think more clearly.Analysis is mostly about making calculus mathematically precise and perfect
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s_chanch
01/16/2022 05:07:14
wel
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s_chanch
01/16/2022 05:08:53
if c is an inflexion point of f (that goes from concave to convex), ∀ δ > 0, ∃ x ∈ (c - δ, c) and y ∈ (c, c + δ) such that x < f(x) and y > f(y)
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s_chanch
01/16/2022 05:08:57
i would define it this way
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s_chanch
01/16/2022 05:09:17
not sure how that helps tho
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sreedev
01/16/2022 05:10:59
Ok,I have a different idea I think.We can show that at local maxima f"(x)<0 and at local minima f"(x)>0
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sreedev
01/16/2022 05:11:56
No,you just solved the problem here I think
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Camilleone
01/16/2022 05:11:59
to be honest, i don't think this is a definition of an inflexion point
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Camilleone
01/16/2022 05:12:15
that is, f''(x) = 0 is _not_ a definition
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Camilleone
01/16/2022 05:12:35
it's a necessary condition, but not sufficient
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s_chanch
01/16/2022 05:15:01
oh and
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s_chanch
01/16/2022 05:15:04
f(x) is well defined
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s_chanch
01/16/2022 05:15:09
????
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s_chanch
01/16/2022 05:15:33
wth?
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sreedev
01/16/2022 05:19:48
Nvm, I'm not really sure whether it will work
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s_chanch
01/16/2022 05:23:23
what other conditions are there
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sreedev
01/16/2022 05:23:53
This is easy to show I think.In local maxima we know that f first increases and then decreases.Thus f'(x) changes sign from positive to -ve as it passes through c.So we can find a nbd of c were f' is strictly decreasing which implies f"(c)<0
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Camilleone
01/16/2022 05:25:28
the definition is that the curvature changes sign
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sreedev
01/16/2022 05:26:02
Similar idea for local minima as well.So since this is true and here f"(x)=0,it cannot be a local max/min and thus is not an extremum point
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s_chanch
01/16/2022 05:26:10
f”(x) is something that comes out
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s_chanch
01/16/2022 05:26:11
isnt it
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Camilleone
01/16/2022 05:27:49
i think this is equivalent to the existence of a $\varepsilon > 0$ s.t. $f''(x-y)$ and $f''(x+y)$ having opposite signs for $0 < y < \varepsilon$
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TeXit
01/16/2022 05:28:16
**Camilleone**
https://cdn.discordapp.c…697925898261.png
569167697925898261.png
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sreedev
01/16/2022 05:28:23
Like the curvature from differential geometry?
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Camilleone
01/16/2022 05:28:41
except specialised to curves, yes
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s_chanch
01/16/2022 05:28:46
bruh
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Camilleone
01/16/2022 05:28:54
if i'm not wrong, the signed curvature of a graph has the same sign as f''
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Camilleone
01/16/2022 05:29:05
so you can just say it's f'' instead
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s_chanch
01/16/2022 05:30:20
im considering convex function as ‘for any segment on the curve, all points except the endpoints are above the curve’
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s_chanch
01/16/2022 05:30:27
then we convert it into algebraci defn
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s_chanch
01/16/2022 05:30:29
then do math on jt
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sreedev
01/16/2022 05:38:52
f"(x)=0 is a consequence of this definition I think,by IVT.So it is a neccesarry condition If we assume this definition
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sreedev
01/16/2022 05:45:58
I'm sorry,I think I have to think about this.I will let you know later.
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s_chanch
01/16/2022 05:48:52
@Helpers
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nyann
01/16/2022 06:41:26
well
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nyann
01/16/2022 06:41:50
first of all you have a wrong statement here I believe
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nyann
01/16/2022 06:42:02
consider the counterexample of x^4 at x=0
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nyann
01/16/2022 06:42:31
also you can generalize this under the higher derivative test
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nyann
01/16/2022 06:42:37
the proof of which requires taylor's theorem
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sreedev
01/16/2022 06:43:09
Oh,right!. Here there is a local minima at 0 but f"(0)=0
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nyann
01/16/2022 06:43:35
so I could write it out here :/ but then that would first rquire the proof of taylor's theorem
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nyann
01/16/2022 06:44:07
so I think its better you look up the proof for higher derivative test and go down the rabbithole if you like
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s_chanch
01/16/2022 07:14:29
taylor’s theorem??
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s_chanch
01/16/2022 07:15:50
so how do we very whether a point is inflexion point lol
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s_chanch
01/16/2022 07:15:54
find f’’’(x)???
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sreedev
01/16/2022 07:15:59
I think,I got a proof using the other definition
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sreedev
01/16/2022 07:17:00
I showed that derivative won't change sign implying it is not an extremum point
https://cdn.discordapp.c…2335309088_1.jpg
1642335309088_1.jpg
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sreedev
01/16/2022 07:24:41
We use MVT to find sign's of derivative.If I didn't made any calculation mistakes,f' have the same sign in the entire interval if f changes from concave to convex or convex to concave.
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sreedev
01/16/2022 07:25:33
Which will imply those are not max or min as at max and min,derivative changes sign
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s_chanch
01/16/2022 07:29:01
ok so apparently
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s_chanch
01/16/2022 07:29:20
this is called an undulation point?
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s_chanch
01/16/2022 07:29:25
haven’t heard of it
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s_chanch
01/16/2022 07:29:27
not sure the use
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nyann
01/16/2022 07:30:12
those names come from diff geo not very common in analysis or calc pretty sure
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nyann
01/16/2022 07:30:32
changes sign is a better way of putting it
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nyann
01/16/2022 07:30:48
if f'(x) changes sign at a point then it is a minima
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nyann
01/16/2022 07:30:53
which is the extremum test
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s_chanch
01/16/2022 07:31:36
hsjdksdknffkndsjncdmndfc
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nyann
01/16/2022 07:35:04
ok so again this one uses mean value theorem
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nyann
01/16/2022 07:38:58
I am done trying to latex it live so let me find some source -.-
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205,438
help-1
9aac582ad5fb4c04bb45b087ffa94d43
373,549,242,754,793,500
nyann
01/16/2022 07:40:18
ok I found this
https://cdn.discordapp.c…1362/unknown.png
unknown.png
0
1
null
null
null
null
null
205,438
help-1
9aac582ad5fb4c04bb45b087ffa94d43
701,895,792,763,600,900
s_chanch
01/16/2022 07:41:23
this is what i was investigating here this afternoon lol
1
0
null
null
null
null
null
205,438
help-1
9aac582ad5fb4c04bb45b087ffa94d43
373,549,242,754,793,500
nyann
01/16/2022 07:41:45
and you can see what sreedev has sent is very similar
0
1
null
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null
205,438
help-1
9aac582ad5fb4c04bb45b087ffa94d43
701,895,792,763,600,900
s_chanch
01/16/2022 07:41:53
where
1
0
null
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null
205,438
help-1
9aac582ad5fb4c04bb45b087ffa94d43
373,549,242,754,793,500
nyann
01/16/2022 07:42:02
https://discord.com/channels/268882317391429632/269573202018041856/932247065860386856
0
1
null
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null
null
null
205,438
help-1
9aac582ad5fb4c04bb45b087ffa94d43
373,549,242,754,793,500
nyann
01/16/2022 07:42:21
except it talks about second derivative at that point changing signs
0
1
510,789,298,321,096,700
TeXit
0796
TeXit
true
205,438
help-1
9aac582ad5fb4c04bb45b087ffa94d43
373,549,242,754,793,500
nyann
01/16/2022 07:43:04
and from there he uses this
0
1
null
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null
205,438
help-1
9aac582ad5fb4c04bb45b087ffa94d43
701,895,792,763,600,900
s_chanch
01/16/2022 07:43:55
isn’t just the first derivative test
1
0
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205,438
help-1
9aac582ad5fb4c04bb45b087ffa94d43
701,895,792,763,600,900
s_chanch
01/16/2022 07:44:03
just we’re doing it on derivatives
1
0
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205,438
help-1
9aac582ad5fb4c04bb45b087ffa94d43
701,895,792,763,600,900
s_chanch
01/16/2022 07:44:18
so let f’(x) = g(x)
1
0
null
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null
205,438
help-1
9aac582ad5fb4c04bb45b087ffa94d43
701,895,792,763,600,900
s_chanch
01/16/2022 07:44:24
we’re investigating g’(x)
1
0
null
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null
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null
205,438
help-1
9aac582ad5fb4c04bb45b087ffa94d43
701,895,792,763,600,900
s_chanch
01/16/2022 07:44:44
and nothing more
1
0
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null
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null
205,438
help-1
9aac582ad5fb4c04bb45b087ffa94d43
701,895,792,763,600,900
s_chanch
01/16/2022 07:44:52
@sreedev is this right
1
0
null
null
null
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null
205,438
help-1
9aac582ad5fb4c04bb45b087ffa94d43
701,895,792,763,600,900
s_chanch
01/16/2022 07:45:03
(you should get advanced role and helper role)
1
0
null
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null
205,438
help-1
9aac582ad5fb4c04bb45b087ffa94d43
373,549,242,754,793,500
nyann
01/16/2022 07:46:06
yeah in a way yes
0
1
null
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null
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null
205,438
help-1
9aac582ad5fb4c04bb45b087ffa94d43
373,549,242,754,793,500
nyann
01/16/2022 07:46:34
if you show g(x) has extremum at c then you can show it does not change signs yes?
0
1
null
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null
205,438
help-1
9aac582ad5fb4c04bb45b087ffa94d43
701,895,792,763,600,900
s_chanch
01/16/2022 08:19:11
also
1
0
null
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null
205,438
help-1
9aac582ad5fb4c04bb45b087ffa94d43
701,895,792,763,600,900
s_chanch
01/16/2022 08:19:21
does the third derivative tell us something?
1
0
null
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null
205,438
help-1
9aac582ad5fb4c04bb45b087ffa94d43
701,895,792,763,600,900
s_chanch
01/16/2022 08:19:31
about function
1
0
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null
205,438
help-1
9aac582ad5fb4c04bb45b087ffa94d43
701,895,792,763,600,900
s_chanch
01/16/2022 08:33:42
.close
1
0
null
null
null
null
null
205,439
help-1
c9dca0024e394d258ac4b5f9f3720a83
774,994,557,859,135,500
Legendang
01/16/2022 08:36:03
https://cdn.discordapp.c…20116_212721.jpg
IMG_20220116_212721.jpg
1
0
null
null
null
null
null
205,439
help-1
c9dca0024e394d258ac4b5f9f3720a83
774,994,557,859,135,500
Legendang
01/16/2022 08:36:10
Can anyone help me with this question?
1
0
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null
205,440
help-1
cd054d24808d4042b01ddee588449ade
710,883,233,595,654,100
R31
01/16/2022 09:33:45
https://cdn.discordapp.c…1626/unknown.png
unknown.png
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