Solution
= Solution
For a unit direction $\mathbf v$, the <Cauchy-Schwarz inequality> gives $\nabla f\cdot\mathbf v\leq|\nabla f|$, with equality in the <gradient> direction. Its norm here is $\sqrt{4+9+1}=\sqrt{14}$, so the direction of fastest increase is
$$
\boxed{\mathbf v_{\max}=\frac{(2,-3,1)}{\sqrt{14}}}.
$$
The maximal <directional derivative> itself is $\sqrt{14}$.