I have a question that I cannot manage to get around. I need to answer the following:
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.
Now, I understand the property that states that concavity is preserved by negative linear combinations. So my first question is, is the contrary always true? That is, that concavity is not respected by positive linear combinations?
If this is not the case always, my main issue here is how to get an example that clearly shows the above for ANY positive linear combination. I imagine this would require an example and a proof but I cannot work around this and to the best of my knowledge this is not covered by another thread.
Thanks in advance, any help would be useful
The function $x\mapsto \sqrt |x|$ is quasi-convex. Let me show that the function $$ f(x) = a \sqrt{|x-1|} + b \sqrt{|x+1|} $$ is not quasi-convex for all $a,b>0$.
The points $x=-1$ and $x=1$ are local minima of $f$. On the interval $(-1,1)$ the function $f$ reduces to $$ f(x) = a \sqrt{1-x} + b \sqrt{x+1}, $$ which is a strictly concave function. Hence, $f$ has a local maximum $x^*\in (-1,1)$ 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 $$ \left\{x : f(x) \le \frac{f(x^*)+\max (f(-1),f(1))}2\right\} $$ contains $-1,1$ but not $x^*$. Hence this level set is not convex, and $f$ is not quasi-convex.
Note that $-f$ is not quasi-concave, but is the sum of two quasi-concave functions. So is is the example you are looking for.