Prove that the function $f(x_1 , x_2) = x_1 x_2$ is quasi-concave on $S = \mathbb{R}_{++}^2$
I started with the definition of quasi-concavity:
$$ f( \lambda x + (1-\lambda) y) \geq \min \left( f(x), f(y) \right) $$
Case 1: $$(λ * x_1 + (1-λ) * y_1) * (λ * x_2 + (1-λ) * y_2) ≥ x_1 * x_2$$ Case 2: $$(λ * x_1 + (1-λ) * y_1) * (λ * x_2 + (1-λ) * y_2) ≥ y_1 * y_2$$
But calculating it for the first case I get: $$x_1 * x_2 * (λ^2-1) + (1-λ) * λ * y_1 * x_2 + (1-λ) * λ * x_1 * y_2 + (1-λ)^2 * y_1 * y_2 ≥ 0$$
And I think now I have to proof that for all positive $x_1,x_2,y_1,y_2$ the equation is satisfied. But is that even possible? I think it depends on how one chooses λ.