How to find a affine function$\ \leq \ $convex function? let $f:\mathbb R^n \rightarrow \mathbb R$ be a convex function.

for arbitrary $p\in \mathbb R^n$, how can I find an affine function $\ell_p(x)$ such that

$$\ell_p(x)\leq f(x)\ \text{ and }\ \ell_p(p)=f(p)?$$

I could find it for $n=1$, but I don't know what to do when $n\geq2$.
Taking the axial sections we arrive at convex functions of one variable. A convex function of one variable admits one-sided derivatives at every interior point of a domain. That is why all one-sided parial derivatives do exist at $p$. It is enough to take a plane described by @Jack D'Aurizio with, say, left-hand-side partial derivatives at $p$.

EDIT

As @JonnMa said, it is not enough. See my comment below.
Taking the axial sections we arrive at convex functions of one variable. A convex function of one variable admits one-sided derivatives at every interior point of a domain. That is why all one-sided parial derivatives do exist at $p$. It is enough to take a plane described by @Jack D'Aurizio with, say, left-hand-side partial derivatives at $p$.

EDIT

As @JonnMa said, it is not enough. See my comment below.