how do we find the direction(s) from a given f(x,y) when directional derivative =0?
By definition, the directional derivative Du(f) = <f_x,f_y>*<u(x),u(y)> where f_x represents the partial derivative of f with respect to x, <u(x), u(y)> is the UNIT vector of the direction in which you are taking the directional derivative, and * is the dot product.
So Du(f) = 0 ==> <f_x,f_y>*<u(x),u(y)> = 0.
==>(1) f_x u(x) + f_y u(y) = 0 by definition of dot product.
Since we’re looking for the unit vector (which is unique),
(2) u(x)^2 + u(y)^2 = 1 (definition of unit vector).
So at this point you can solve for u(x) or u(y) in equation (1) and then substitute this into equation (2) to solve for one of the components of the u vector. Then solve for the other value using eq. (2). Once you have u(x) and u(y), the direction is