21. Given the function $f ( x ) = - 2 ( x + a ) \ln x + x ^ { 2 } - 2 a x - 2 a ^ { 2 } + a$, where $a > 0$.\\
(1) Let $g ( x )$ be the derivative of $f ( x )$. Discuss the monotonicity of $g ( x )$;\\
(2) Prove: there exists $a \in ( 0, 1 )$ such that $f ( x ) \geq 0$ holds on the interval $(1, + \infty)$, and $f ( x ) = 0$ has a unique solution in $(1, + \infty)$.