answered
2014-07-11 08:48:24 -0600
If you go through the process of computing the directional derivative, you should have the gradient of $f$ dotted with $\hat u$:
$$D_{\hat u} = \nabla f \cdot \hat u$$
Here, $\nabla f = \langle 2, 1 \rangle$, and let's say that $\hat u = \langle u_1, u_2 \rangle$. Then the directional derivative is
$$\langle 2,1 \rangle \cdot \langle u_1, u_2 \rangle$$
$$= 2u_1 + u_2 $$
We want the direction with the derivative equal to one, so we should set this whole thing equal to one:
$$2u_1 + u_2 = 1$$
However, the vector also has to be of unit-length, so we know that $u_1^2 + u_2^2 =1$. Now we have two equations and two unknowns; the solutions to this system should give you the components of $\hat u$ that work. I found another solution on Mathematica, but I'm not sure how you could solve this easily by hand.