# Proof on quasi-concavity and concavity

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?

If $$f_i$$ is quasiconcave then, $$\{u,v\}\in$$ domain of f for $$0, and for $$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

• What does the "*duplicated question" refer to? Duplicated from where? Oct 15, 2022 at 23:27

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.