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Let $g^1\colon\mathbb{R}\to\mathbb{R}$ and $g^2\colon\mathbb{R}\to\mathbb{R}$ be concave functions, and let $f\colon\mathbb{R}\to\mathbb{R}$ be a non-decreasing function (i.e., $f(x)≥f(y)$ whenever $x≥y$).

Let $h\colon\mathbb{R}^2\to\mathbb{R}$ be defined by: $$h(x_1,x_2)=f(g^1(x_1)+g^2(x_2)).$$

How do I prove that $h$ is quasi-concave?

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    Prove that for all $x_1,x_2,x_1',x_2'$ and $0\leq\alpha\leq 1$: $h(\alpha(x_1,x_2)+(1-\alpha)(x_1',x_2'))\geq\min \{f(x_1,x_2),f(x_1',x_2')\}$.2011-10-09

2 Answers 2