Question

Find a unit vector for which the directional derivative is zero

Answer 1

The directional derivative is zero in the directions of u = 〈−1, −1〉/ √2 and u = 〈1, 1〉/ √2. If the gradient vector of z = f(x, y) is zero at a point, then the level curve of f may not be what we would normally call a “curve” or, if it is a curve it might not have a tangent line at the point.