$K\subset \mathbb{R}^k$- convex, $f$ differentiable on $K$, $(a_1,\ldots, a_k)$ - non-zero sequence. We know that
$$\forall_k\sum\limits^{k}_{j=1}a_j\frac{\partial f}{\partial x_j}\leq0$$
Show that $f$ has no local proper extremum inside $K$
Basically I don't know if I understand what can I get from convexity. We have a proper local maximum at $x_0$ if there exists an open neighborhood $U$ of the point $x_0$ such that for every $x$ in the set $U$, except $x_0$, the condition $f(x) < f (x_0)$ is satisfied. Is the intuition that f does not increase in either direction? Any hints?
