Let $\displaystyle I$ be an open interval where $\displaystyle a<x<b\in I$. Also, let $\displaystyle f:I\rightarrow \mathbb{R}$ be differentiable.
Prove $\displaystyle f$ is convex if and only if $\displaystyle f(x)\geq f(a)+f'(a)(x-a)$ for all $\displaystyle a,x\in I$.

I have that the definition of convex is $\displaystyle f(x)\leq L(x)$ where $\displaystyle L(x)=f(a)+\frac{f(b)-f(a)}{b-a}(x-a)$

I have tried to switch it around algebraically but I can't seem to figure it out. I'd appreciate any help.