Extremum, directional derivatives If a function $f: \mathbb{R}^n \to \mathbb{R}$ has all the directional derivatives zero at a point $c \in \mathbb{R}^n$, does it imply that $f$ has an extremum at $c \in \mathbb{R}$? If so, how do I prove it?
No, it doesn't because it is only a necessary condition for a local extremum. You should check your point $c$ for the fulfillment of a sufficient condition for a local extremum.
No, it doesn't because it is only a necessary condition for a local extremum. You should check your point $c$ for the fulfillment of a sufficient condition for a local extremum.