Let $f:\mathbb{R}\to\mathbb{R}$ be defined as $f(x)=\begin{cases} cx^{2}+ax+b, & x<0,\\ bx^{2}+cx+a, & 0\le x<2,\\ ax^{2}+bx+c, & x\ge 2, \end{cases}$ where $a,b,c$ are positive real numbers. Which of the following statements is correct, under the assumption that $f$ is continuous?