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.

Question

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.

scranna0o · Accepted Answer

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  &amp;gt;  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    ∗    )  &amp;gt;  max  (  f  (  −  1  )  ,  f  (  1  )  ).which is a strictly concave function. Hence,   f has a local maximum       x    ∗    ∈  (  −  1  ,  1  ) with   f  (      x    ∗    )  &amp;gt;  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.

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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