descent versus ascent direction Suppose $f$ is a function from $\mathbb{R}^n \to \mathbb{R}$ If $\langle\nabla f(x), p \rangle$ =0, is $p$ an ascent or descent direction for $f$ at point $x$?

Note that $\langle\
abla f(x),p\rangle=\frac{\partial}{\partial t}|_{t=0}f(x+tp)$ is the derivative of the restriction of $f$ to the line $x+tp$. Hence being zero means $t=0$ (hence $x$) is a critical point of that restriction. Thus it is endeed a candidate to be ascent or descent direction, but it can be either or none. One needs another dervative to decide.