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
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