Prove that function do not have infinum and supremum inside its domain

Question

$A \subseteq \Re^n$ convex and compact.

$f: A\rightarrow \Re$ continuous and differentiable.

There are numbers $a_{1},\dots,a_{n}$ and not all of them equal zero.

For all $x \in IntA$ it is true that: $ \sum_{i=1}^{n} a_{i} \frac{\partial f}{\partial x_{i}} \geq 0$

Prove that $f$ reaches its supremum and infinum on the edges of $A$

I do not have any ideas worth mentioning there.

Answer 1

Hint: Suppose $U$ is open, convex and bounded, $f:\overline U \to \mathbb R$ is continuous, and $f$ is differentiable on $U.$ Set $a = (a_1,\dots,a_n).$ Let $x_0 \in U.$ Show $t\to f(x_0 + ta)$ is increasing.