Proof on quasi-concavity and concavity

Question

Let $f_1,…, f_m$ be quasiconcave functions on a convex subset X on $IR^n$. Let $p_1,…, p_m$ be nonnegative real numbers.

Then, how can I show that $\sum_{i=1}^mp_if_i $ is also quasi concave function?

Similarly, Let $g_1,…, g_m$ be concave functions on a convex subset X on $IR^n$. Let $p_1,…, p_m$ be nonnegative real numbers.

Then, how can I show that $\sum_{i=1}^mp_ig_i $ is also concave function?

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If $f_i$ is quasiconcave then, $\{u,v\}\in$ domain of f for $0<a<1$, and for $f_i(u)> f_i(v)$ $f_i(au+(1-a)v)\ge f_i(v)$

Similarly, for concavitiy, $g_i(au+(1-a)v)\ge ag_i(u)+(1-a)g_i(v)$

And then, how can I proceed the proof?

Thank you. All helps will be appreciated.

*duplicated question

Answer 1

Let $f_1:\mathbb{R}\to\mathbb{R}$ be given by $f_1(x)=0$ for $x<0$ and $f_1(x)=x^2$ for $x\geq 0$. Also, let $f_2:\mathbb{R}\to\mathbb{R}$ be given by $f_2(x)=x^2$ for $x<0$ and $f_2(x)=0$ for $x\geq 0$. Both $f_1$ and $f_2$ are quasi-concave but their sum $f_1+f_2$ given by $(f_1+f_2)(x)=x^2$ for all $x\in\mathbb{R}$ is not quasi-concave.

However, the result does hold true for concave functions. Show that if $p\geq 0$ and $f$ is concave, then $pf$ is concave too. Then show that if $f$ and $g$ are concave, then $f+g$ is concave too. The rest follows by induction.