We are given $f \in \mathcal{C}^1(\mathbb{R})$, convex, admitting a minimizer $x_* \in \mathbb{R}$, with $f'$ being $L$-Lipschitzian, and $0 < \tau < 2/L$. Show that $0 \leq f(x) - f(x_*) \leq |x - x_*||f'(x)|$ for all $x \in \mathbb{R}$.
We are given $f \in \mathcal{C}^1(\mathbb{R})$, convex, admitting a minimizer $x_* \in \mathbb{R}$, with $f'$ being $L$-Lipschitzian, and $0 < \tau < 2/L$.\\
Show that $0 \leq f(x) - f(x_*) \leq |x - x_*||f'(x)|$ for all $x \in \mathbb{R}$.