A Unique Zero in an Interval

Let $f(x)=-2(x+a)\ln x+x^2-2ax-2a^2+a$, where $a>0$.

1. Let $g(x)$ be the derivative of $f(x)$. Discuss the monotonicity of $g(x)$. 2. Prove that there exists $a\in(0,1)$ such that $f(x)\geqslant 0$ holds throughout $(1,+\infty)$, and $f(x)=0$ has a unique solution in $(1,+\infty)$.