We consider a convex function $f \in \mathcal{C}(\mathbb{R})$, admitting a minimizer $x_* \in \mathbb{R}$. Let also $\tau > 0$.\\
Show that the function $F_{x_0}(x) := \frac{1}{2}|x - x_0|^2 + \tau f(x)$ admits a unique minimizer on $\mathbb{R}$, which we will denote $p_f(x_0)$.\\
Hint: We may consider minimizers $x_1$ and $x_2$ of $F_{x_0}$, and note that
$$\left|\frac{1}{2}(x_1 + x_2) - x_0\right|^2 < \frac{1}{2}|x_1 - x_0|^2 + \frac{1}{2}|x_2 - x_0|^2 \quad \text{if } x_1 \neq x_2$$