Problem with convex curve and Jensen inequality

given a convex curve function f(x),

Jensen inequality states:

$\displaystyle E[ f(x) ] \geq f( E[x] )$

Preliminary:

Let l(x) be linear function below f(x). Since a convex function is the maximum of all linear functions that lie below it:

$\displaystyle f(x) \geq max\{l(x)\}$

>>> this equation I understand.

On the other hand, because f(x) is convex, there is always a value x for which f(x) = l(x), where l(x) is the tangent (or support line) to f(x).

So,

$\displaystyle f(x) \leq max\{l(x)\}$

>>> I don't understand at all.

So f(x) = max{l(x)}.

This is a relationship we are going to use to prove Jensen's inequality.

Can anyone explain the problem I have with the 2nd inequality first before I post the rest of the solution (which I can follow.) ?

Thanks.