We consider a convex function $f \in \mathcal{C}(\mathbb{R})$, admitting a minimizer $x_* \in \mathbb{R}$, and $\tau > 0$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $x_{n+1} := p_f(x_n)$. We have established that $p_f(x_{**}) = x_{**}$ for some $x_{**} \in \mathbb{R}$. Conclude that $x_{**}$ is a minimizer of $f$, and that $x_n \rightarrow x_{**}$ as $n \rightarrow \infty$.

Let $f : [ 0,1 ] \rightarrow \mathbb { R }$ be a continuous function which is differentiable on $( 0,1 )$. Prove that either $f$ is a linear function $f ( x ) = a x + b$ or there exists $t \in ( 0,1 )$ such that $| f ( 1 ) - f ( 0 ) | < \left| f ^ { \prime } ( t ) \right|$.