Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a real valued quasi-convex function, and $g: \mathbb{R} \rightarrow \mathbb{R}$ be a linear function.
Is $(f+g)(x) = f(x) + g(x)$ a quasi-convex function?
Thank you.
2026-03-30 17:13:23.1774890803
Quasiconvex plus a linear function
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No, consider $f(x) = x^3$ (quasiconvex) and $g(x)=-x$ (linear). Then $f+g$ is not quasiconvex, since $\{x : (f+g)(x) \leq 0\} = (-\infty,-1] \cup [0,1]$.