RO-NMO 2017 Problem 94

Let be two natural numbers $b>a>0$ and a function $f:\mathbb{R}\longrightarrow\mathbb{R}$ having the following property. $$f\left( x^2+ay\right)\ge f\left( x^2+by\right) , \forall x,y\in\mathbb{R}$$ 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.