Does there always exist $x_0 \in \mathbb{R}$ such that $f(x_0) + f'(x_0)(x - x_0) + \frac{f''(x_0)}{2}(x - x_0)^2 \ge 0$ for all $x \in \mathbb{R}$?

Question

Let $f: \mathbb{R} \to (0, \infty)$ be two times differentiable function. Does there always exist $x_0 \in \mathbb{R}$ such that $f(x_0) + f'(x_0)(x - x_0) + \frac{f''(x_0)}{2}(x - x_0)^2 \ge 0$ for all $x \in \mathbb{R}$?