Spring 2017, problem 40
Let $f: \mathbb R\longrightarrow\mathbb R^{+}$ be a non-decreasing function. Show that there is a point $a\in\mathbb R$ that $$f\left(a+\frac1{f(a)}\right) \lt 2f(a).$$
Let $f: \mathbb R\longrightarrow\mathbb R^{+}$ be a non-decreasing function. Show that there is a point $a\in\mathbb R$ that $$f\left(a+\frac1{f(a)}\right) \lt 2f(a).$$