(a) Consider the function $f\left(x_{1}, x_{2}\right)=2 x_{1}^{2}+x_{2}^{2}+\alpha x_{1} x_{2}$, where $\alpha$ is a real constant. For what values of $\alpha$ is the function $f$ convex?

(b) In the case $\alpha=-3$, calculate the extremum of $x_{1}^{2}$ on the set of points where $f\left(x_{1}, x_{2}\right)+1=0 .$