Problem

Source: Romanian National Olympiad 2017, grade ix, p. 4

Tags: function, algebra, Find all functions



Let be two natural numbers $ b>a>0 $ and a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following property. $$ f\left( x^2+ay\right)\ge f\left( x^2+by\right) ,\quad\forall x,y\in\mathbb{R} $$ a) Show that $ f(s)\le f(0)\le f(t) , $ for any real numbers $ s<0<t. $ b) Prove that $ f $ is constant on the interval $ (0,\infty ) . $ c) Give an example of a non-monotone such function.