Hi all, I have a simple question:

Suppose $\displaystyle f$ is a continously differentiable convex function on $\displaystyle \mathbb{R}^n$.

Then by definition (1):

$\displaystyle

f(t x_1 + (1-t)x_2) \leq t f(x_1) + (1-t) f(x_2)

$

I've also read that (2):

$\displaystyle

f(x_1) \geq f(x_2) + \nabla f(x_2)[x_1-x_2]

$

Can anybody tell me, if (1) implies (2), and why?