How do you find the critical point(s) of #f(x,y) = (x-y)^2#?
1 Answer
The critical points of a two-variables functions are to be found using the gradient.
The gradient is a vector which has dimension equal to the number of variables: in this case, 2.
The coordinates of the gradient are the derivatives with respect to each variable the function depends on. In this case, the vector will be a 2-dimensional vector, where the first coordinate is the derivative with respect to
Note that deriving with respect to a variable means to consider the other as a constant.
Now, expand the square in the definition of
Deriving with respect to
Deriving with respect to
Now, critical points of a functions are the points in which the gradient equals the zero vector. This happens if the following system is solved:
Both equations yield the line