Convex function: sufficient condition $\forall a>b,f(a)\geq f(b)+f'(b)(a-b)$ $\implies$ $f$ convex?

Question

Suppose $f\in C^1$, does the condition that $\forall a>b,f(a)\geq f(b)+f'(b)(a-b)$ imply $f$ is convex?

We know that $f$ is convex if and only if $\forall a,b, \ f(a)\geq f(b)+f'(b)(a-b)$. But I want to know whether the condition $\forall a>b,f(a)\geq f(b)+f'(b)(a-b)$ is sufficient or not.