Give an example to 2 quasi-concave functions on an interval such that any positive linear combination of these two functions is not quasi-concave.

Aryanna Blake 2022-10-23 Answered
Give an example to 2 quasi-concave functions on an interval such that any positive linear combination of these two functions is not quasi-concave.
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scranna0o
Answered 2022-10-24 Author has 16 answers
The function x | x | is quasi-convex. Let me show that the function
f ( x ) = a | x 1 | + b | x + 1 |
is not quasi-convex for all a , b > 0.
The points x = 1 and x = 1 are local minima of ( 1 , 1 ). On the interval ( 1 , 1 )the function ( 1 , 1 ) reduces to
f ( x ) = a 1 x + b x + 1 ,
which is a strictly concave function. Hence, f has a local maximum x ( 1 , 1 ) with f ( x ) > max ( f ( 1 ) , f ( 1 ) ).
which is a strictly concave function. Hence, f has a local maximum x ( 1 , 1 ) with f ( x ) > max ( f ( 1 ) , f ( 1 ) ).
Now let me choose a sub-level set that contains 1 and 1 but not x . Then the sub-level set
{ x : f ( x ) f ( x ) + max ( f ( 1 ) , f ( 1 ) ) 2 }
contains 1 , 1 but not x . Hence this level set is not convex, and f is not quasi-convex.
Note that f f is not quasi-concave, but is the sum of two quasi-concave functions.
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