Fix compact intervals $X, P \subseteq \mathbb{R}$.

Let $f_n : X \times P \to \mathbb{R}$ be a sequence of $C^2$ functions converging uniformly to a $C^2$ function $f$. The first and second derivatives of $f_n$ also converge uniformly to those of $f$.

For any $p \in P$, the function $f(\cdot,p) : X \to \mathbb{R}$ has a unique zero $x^*(p)$, and $f_1(x^*(p),p) \neq 0$. Therefore there is a sequence $x_n^* (p) \to x^*(p)$ such that $f_n(x_n^*(p),p)=0$ for large $n$.

When can we also say that $x_n^*(p) \to x(p)$ uniformly in $p$? Is there a standard reference for such results?