Convex and concave functions in Real numbers -- Basic 1
by adityaguharoy, Mar 1, 2018, 1:44 PM
Convex functions
Let
be a function, and let 
be two real numbers with 
. Then we say that 
is a convex function on the interval ![$[a,b]$](//latex.artofproblemsolving.com/8/e/c/8ecbd1ba3da8f2adef66a63f2ab32c47e63fa734.png)
if and only if the following is true :
Given any![$t \in [0,1]$](//latex.artofproblemsolving.com/6/7/3/6735b925696750e153b8d293780a7b620449b778.png)
, and , any ![$x_1 , x_2 \in [a,b]$](//latex.artofproblemsolving.com/b/3/a/b3a60e6a3e809b381d3ae396425baf202335a943.png)
then,
And we say that is strictly convex on if the above inequality is strict whenever 
and 
.
Concave functions
Let be a function, and let be two real numbers with . Then we say that is a concave function on the interval if and only if the following is true :
Given any , and , any then,
And we say that is strictly concave on if the above inequality is strict whenever and .
Let
be a function, and let 
be two real numbers with 
. Then we say that 
is a convex function on the interval ![$[a,b]$](http://latex.artofproblemsolving.com/8/e/c/8ecbd1ba3da8f2adef66a63f2ab32c47e63fa734.png)
if and only if the following is true :Given any
![$t \in [0,1]$](http://latex.artofproblemsolving.com/6/7/3/6735b925696750e153b8d293780a7b620449b778.png)
, and , any ![$x_1 , x_2 \in [a,b]$](http://latex.artofproblemsolving.com/b/3/a/b3a60e6a3e809b381d3ae396425baf202335a943.png)
then,
And we say that is strictly convex on if the above inequality is strict whenever 
and 
.Concave functions
Let
be a function, and let be two real numbers with . Then we say that is a concave function on the interval if and only if the following is true :Given any
, and , any then,
And we say that is strictly concave on if the above inequality is strict whenever and .
Let 
be a function and 
is concave on 
is defined by 
)
Let 
(the double derivative of 
Derive a version of 
(as above) for concave functions.This post has been edited 2 times. Last edited by adityaguharoy, Mar 4, 2018, 7:25 AM
Relaxing the to-be satisfied conditions for Squeezing Monotone sequences
by adityaguharoy, Feb 28, 2018, 4:09 PM
The Sandwich theorem (or Squeeze Principle ) for sequences of real numbers says that :
If
, 
and 
be three sequences of real numbers which obeys all the following :
Both the sequences and converges
Then, must converge, and also obey
Forgetting the third condition (as given above) can lead us to errorenous results.
For example : If we define as 
and, we define as 
, and we define the sequence as 
, then 
and both the sequences and converges, but however, diverges.
But, it seems we can relax the conditions for some special type of sequences. Let us see how, we can relax it for monotone sequences.
Note, that the Monotone Convergence Theorem (of Bolzano Weirestrass) says that :
Every monotone bounded sequence of real numbers must converge.
Thus, if , , and , be three monotone sequences such that
Both the sequences , and , converges
Then,
The sequence must converge , and ,
If
, 
and 
be three sequences of real numbers which obeys all the following :
Both the sequences and converges 
Then,
must converge, and also obey
Forgetting the third condition (as given above) can lead us to errorenous results.For example : If we define
as 
and, we define as 
, and we define the sequence as 
, then 
and both the sequences and converges, but however, diverges.But, it seems we can relax the conditions for some special type of sequences. Let us see how, we can relax it for monotone sequences.
Note, that the Monotone Convergence Theorem (of Bolzano Weirestrass) says that :
Every monotone bounded sequence of real numbers must converge.
Thus, if
, , and , be three monotone sequences such that 
Both the sequences , and , convergesThen,
The sequence
must converge , and ,
This post has been edited 1 time. Last edited by adityaguharoy, Mar 1, 2018, 1:41 PM
1. Algebra and sigma algebra on a set
by adityaguharoy, Feb 28, 2018, 3:28 PM
be an arbitrary set. Then a collection 
for each 
the set 
for every finite sequence 
of elements each 
the union 
of elements each 
,Let
be an infinite set. And let 
be the collection of all subsets 
of such that either or 
is finite. Prove that is an algebra on , but not a sigma algebra on .
then 
to 
)
be the union 
then 
is an infinite set. Thus, A related exercise :
Let
be an . Let be the collection of all subsets of such that either is countable or 
is countable.Is
a 
algebra on ?This post has been edited 2 times. Last edited by adityaguharoy, Feb 28, 2018, 4:16 PM
Broken Stick Problem with 3 pieces forming a triangle
by adityaguharoy, Jan 22, 2018, 11:50 AM
Let there be a stick of length 
unit. We break it at two random points to get three pieces of the stick. Find the probability that these pieces form a triangle.
unit. We break it at two random points to get three pieces of the stick. Find the probability that these pieces form a triangle.
inside an equilateral triangle 
such that the distances between 
,
and 
form a triangle.
and then using some Euclidean geometry this area 
.Ramsey problem: 3 people in 6 pairwise knowing or strangers
by adityaguharoy, Jan 19, 2018, 11:57 AM
Given any collection of 
people. Prove that there is either a collection of 
people among them such that they know each other pairwise, or there is a collection of people among them who are pairwise strangers.
people. Prove that there is either a collection of 
people among them such that they know each other pairwise, or there is a collection of people among them who are pairwise strangers.
.
.
is a set of 
and 
know each other.
is a set of 
.Happy New Year 2018 wishes
by adityaguharoy, Jan 3, 2018, 3:24 AM
Happy New Year to everyone.
seems interesting and fun.Attachments:
Christmas is coming -- Advance wishes
by adityaguharoy, Dec 20, 2017, 4:15 AM
Only 
days left for Christmas !! 
days left for Christmas !! Attachments:
The oldest, shortest words — "yes" and "no" — are those which require the most thought.
April 2021
February 2021
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February 2018
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