IMC 2009

Let $A,B,C$ be real square matrices of the same order, and suppose $A$ is invertible. Prove that $$(A-B)C=BA^{-1}\implies C(A-B)=A^{-1}B$$

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