EDB — 18H

Exercises

1. [18H]Show that for a convex function \(f:(a,b)→ℝ\) there are only three possibilities:

   • \(f\) is strictly increasing

   • \(f\) is strictly decreasing

   • There are two values \(l_-≤ l_+\) such that \(f\) is strictly increasing in \([l_+,b)\), \(f\) is strictly decreasing in \((a,l_-]\), and the interval \([l_-,l_+]\) are all minimum points of \(f\);

   If also \(f\) is strictly convex then there is at most only one minimum point.