= Solution
The <gradient> is $\nabla f=(8x,2y)$, so $\nabla f(1,1)=(8,2)$. The supplied direction is not a <unit vector>; normalize it to $\hat{\mathbf u}=(1,2)/\sqrt5$. Its <directional derivative> is
$$
\boxed{D_{\hat{\mathbf u}}f=(8,2)\cdot\frac{(1,2)}{\sqrt5}=\frac{12}{\sqrt5}.}
$$
The direction of greatest decrease is opposite to the gradient:
$$
\boxed{\hat{\mathbf n}_{\rm decrease}=-\frac{(8,2)}{\sqrt{68}}=-\frac{(4,1)}{\sqrt{17}},\qquad D_{\hat{\mathbf n}_{\rm decrease}}f=-2\sqrt{17}.}
$$
The <Cauchy-Schwarz inequality> shows that no other unit direction can give a more negative derivative.
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