How to prove a generalised function is quasiconcave? I have a question I have been asked to solve:
Given that $(a_1, a_2,...,a_n)\in R_{++}^n$ and $(x_1, x_2,...,x_n)\in R_{++}^n$, and $A&gt;0, \mu &gt;0, p \neq 0$, if there is a function $f(x)=A(a_1x_1^p+a_2x_2^p+...+a_nx_n^p)^\frac{\mu}{p} \space \space \forall x \in R_{++}^n$, show that function $f$ is a quasiconcave function over $R_{++}^n$ when $p \in (0,1]$.
I understand that to show a function is quasiconcave, it needs to satisfy this definition:
$f:_+^→$ is quasiconcave on $_+^$ if and only if $∀,∈_+^$ and for all $\in(0,1)$
$f(+(1−))≥\{f(),f()\}$
I am wondering how I would use this definition to show that a generalised function $f$ is quasiconcave, or if I should be approaching this from another concept/definition.
Note first that the function $r\mapsto A(r)^{\mu/p}$ is strictly increasing on $\mathbb{R}_+$. When you look at the minimum in the definition of quasi-concavity, you can therefore ignore this part and it suffices to show that the function
$$(x_1,x_2,\ldots,x_n)\mapsto a_1x_1^p+a_2x_2^p+...+a_nx_n^p$$
is quasi-concave. Actually, the function is even concave. The sum of concave functions is concave, so it suffices to show that the function
$$(x_1,x_2,\ldots,x_n)\mapsto a_ix_i^p$$
is concacave for each $i=1,\ldots,n$.