11. (20 points) Given the quadratic functions
$$
y_{1}=a x^{2}+4 x+b \text { and } y_{2}=b x^{2}+4 x+a
$$

both have minimum values, denoted as $m$ and $n$ for $y_{1}$ and $y_{2}$, respectively.
(1) If $m+n=0$, prove that for any real number $x$, $y_{1}+y_{2} \geqslant 0$;
(2) If $m$ and $n$ are both greater than 0, and $m n=2$, let $M$ be the larger of $m$ and $n$. Find the minimum value of $M$.