The exercise says we should verify that
$$f(x,y) = x^3 - xy^2 +y^4 $$
is increasing in the direction of (0,1) at the point (1,1) and decreasing in the direction (-1,0) at the point (1,1).
I calculated $\nabla f_{(1,1)} = (2,2)$, which is the direction of maximum increase. Obviously $-\nabla f_{(1,1)} = (-2,-2)$ is the direction of minimum increase. But how can I find which vectors originating at (1,1) represent increasing or decreasing values of $f$? Do I have to measure the angles they subtend w.r.t. the gradient?
Also, what exactly is the geometrical interpretation of $\| \nabla f_{(1,1)} \|$? Does it measure the rate of increase of $f$ at $(1,1)$ in the direction of $\nabla f_{(1,1)}$?
In general we use the directional derivative $D_u f(x) = \nabla f(x)\cdot u$ to denote the rate of change of $f$ in direction $u \in \mathbb{R^2}$ at point $x$. Here we have found $D_{(0,1)}f(1,1)$ to be $2$ and $D_{(-1,0)}f(1,1)$ to be $-2$.
The norm of grad $f$ at a point denotes the rate of increase of $f$ measured in the norm of choice. E.g. in Pythagorean euclidean norm, it denotes the slope of the function measured in the pythagorean sense.