## Fall 2017, problem 53

Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a two times differentiable function satisfying $f(0) = 1$, $f^\prime(0) = 0$ and for all $x \in [0, \infty)$, it satisfies $$f^{\prime \prime}(x) - 5 f^\prime(x) + 6 f(x) \geq 0.$$ Prove that, for all $x \in [0,\infty)$, $$f(x) \geq 3e^{2x} - 2e^{3x}.$$