Intermediate Value Theorem

For the following question, choose the correct answer from the codes (a), (b), (c) and (d) defined as follows:
Let $f\left(x\right)=\left\{\begin{array}{cc}-a{x}^2-b\left|x\right|-c,& -\alpha \le x<0\\ a{x}^2+b\left|x\right|+c,& 0\le x\le \alpha \end{array}\right.$ where $a,b,c$ are positive and $\alpha >0$, then
Statement I: The equation $f\left(x\right)=0$ has at least one real root for $x\in \left[-\alpha ,\alpha \right].$
because
Statement II: Values of $f\left(-\alpha \right)$ and $f\left(\alpha \right)$ are opposite in sign.