We consider a convex function $f \in \mathcal{C}(\mathbb{R})$, admitting a minimizer $x_* \in \mathbb{R}$, and $\tau > 0$. The operator $p_f$ is defined as the unique minimizer of $F_{x_0}(x) := \frac{1}{2}|x - x_0|^2 + \tau f(x)$.\\
In this question only, we suppose that $f \in \mathcal{C}^1(\mathbb{R})$. Show that $x_1 := p_f(x_0)$ satisfies
$$x_1 = x_0 - \tau f'(x_1)$$