# 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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## Answers (1)

scranna0o
Answered 2022-10-24 Author has 16 answers
The function $x↦\sqrt{|}x|$ is quasi-convex. Let me show that the function
$f\left(x\right)=a\sqrt{|x-1|}+b\sqrt{|x+1|}$
is not quasi-convex for all $a,b>0$.
The points $x=-1$ and $x=1$ are local minima of $\left(-1,1\right)$. On the interval $\left(-1,1\right)$the function $\left(-1,1\right)$ reduces to
$f\left(x\right)=a\sqrt{1-x}+b\sqrt{x+1},$
which is a strictly concave function. Hence, $f$ has a local maximum ${x}^{\ast }\in \left(-1,1\right)$ with $f\left({x}^{\ast }\right)>max\left(f\left(-1\right),f\left(1\right)\right)$.
which is a strictly concave function. Hence, $f$ has a local maximum ${x}^{\ast }\in \left(-1,1\right)$ with $f\left({x}^{\ast }\right)>max\left(f\left(-1\right),f\left(1\right)\right)$.
Now let me choose a sub-level set that contains $1$ and $-1$ but not ${x}^{\ast }$. Then the sub-level set
$\left\{x:f\left(x\right)\le \frac{f\left({x}^{\ast }\right)+max\left(f\left(-1\right),f\left(1\right)\right)}{2}\right\}$
contains $-1,1$ but not ${x}^{\ast }$. 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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