EDB — 18W

[18W] Prerequisites:[18T]↺↻.

Let \(J⊂ ℝ \) be an open nonempty interval, and \(f:J→ℝ\) be a twice differentiable and convex function. Show that the following facts are equivalent:

1. \(f\) is strictly convex,

2. the set \(\{ x∈ J:f''(x)=0\} \) has an empty interior,

3. \(f'\) is monotonic strictly increasing.