We are given $f \in \mathcal{C}^1(\mathbb{R})$ such that $f'$ is $L$-Lipschitzian, with $L > 0$, and $f$ is $\alpha$-convex with $\alpha > 0$, that is $g(x) := f(x) - \frac{1}{2}\alpha x^2$ is a convex function on $\mathbb{R}$.\\
Show that $f(x) \geq f(0) + f'(0)x + \alpha x^2/2$ for all $x \in \mathbb{R}$. Deduce that $f$ admits a minimizer on $\mathbb{R}$.