I read in book (Convex Optimization, boyd) that
quasiconvex (or unimodal) if its domain and all its sublevel sets $S_α = \{x ∈ dom f | f(x) ≤ α\}$, for $α ∈ R$, are convex.
And if and only if $f(x)$ is non-decreasing or non-increasing, $f(x)$ is quasiconvex.
So I wonder if $f(x) = -x^2 + 10$ is quasiconvex, by definition if $α = 5$, sublevel sets are two distinct (-inf, a) and (b, +inf), which is not convex set. But according to second rule, $f(x)$ is not monotonic, which makes it quasiconvex. Where did I made a mistake?
"non-decreasing" means "increasing" (similarly for "non-increasing"): it does not mean "not monotonic". See e.g. Increasing and decreasing.