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I have two functions, $f_1(x)$ and $f_2(x)$. Here $f_1(x) > 0$. The domain is real positive. I have derived that $f'_1(x) < 0$, $f_1''(x) > 0$, $f_1'''(x) < 0$ and so on.. Thus $f_1(x)$ is always positive, strictly decreasing, strictly convex and so on..

Moreover, for $f_2(x)$, we have $f_2(x) > 0$, $f_2'(x) > 0$, $f_2''(x)< 0$, $f'''_2(x) > 0$ and so on.. Thus $f_2(x)$ is always positive, strictly increasing, strictly concave and so on..

What can we say about the quassi-concavity (or convexity/concavity) of $f_1(x) + f_2(x)$?

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Generally we can not say any thing . Let $g : [0 , + \infty) \to \Bbb R$ satisfying same conditions in question as $f_1$ does.

Then by setting $f_1 = 2g$ and $f_2 = -g + f_1(0)$, we have $f_1 + f_2$ is convex (so quasi convex) but not concave.

By setting $f_1 = g$ and $f_2 = -2g + 2f_1 (0)$, we have $f_1 + f_2$ is concave (so quasi concave) but not convex.

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