EDB — 18W

[18W] Prerequisites:[18T]↺↻.

Let $J\subset ℝ$ be an open nonempty interval, and $f:J\to ℝ$ be a twice differentiable and convex function. Show that the following facts are equivalent:

1. $f$ is strictly convex,

2. the set $\{x\in J:f^{″}(x)=0\}$ has an empty interior,

3. $f^{\prime }$ is monotonic strictly increasing.