= Solution
The <gradient> of the <scalar field> is
$$
\nabla f=(2x+9y-2z^2,\;9x,\;-4xz),\qquad
\nabla f(2,4,8)=(-88,18,-64).
$$
For a <unit vector> $\hat{\mathbf n}$, the <directional derivative> is $D_{\hat{\mathbf n}}f=\nabla f\cdot\hat{\mathbf n}$. The <Cauchy-Schwarz inequality> bounds its magnitude by $\|\nabla f\|=2\sqrt{3041}$, with equality precisely in the two directions parallel to the <gradient>. Therefore
$$
\boxed{\hat{\mathbf n}=\pm\frac{(-44,9,-32)}{\sqrt{3041}},
\qquad \max_{\|\hat{\mathbf n}\|=1}|D_{\hat{\mathbf n}}f|=2\sqrt{3041}.}
$$
The plus choice gives the greatest positive <directional derivative>, and the minus choice gives the most negative <directional derivative>. Both maximize its magnitude.
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