Consider the function
f(x, y) = (18x - x^2)(6y - y^2).
Find and classify all critical points of the function. If there are more blanks than critical points,
leave the remaining entries blank. f_x = (18 - 2x)(6y - y^2)
f_y = (18x - x^2)(6 - 2y)
f_{xx} = -2(6y - y^2)
f_{xy} = (18 - 2x)(6 - 2y)
f_{yy} = -2(18x - x^2)
There are several critical points to be listed. List them lexicographically, that is in ascending
order by x-coordinates, and for equal x-coordinates in ascending order by y-coordinates (e.g.,
(1, 1), (2, 1), (2, 3) is a correct order) The critical point with the smallest x-coordinate is
(0, 0) Classification: (local minimum, local maximum, saddle point, cannot be
determined)
The critical point with the next smallest x-coordinate is
(0, 6) Classification: (local minimum, local maximum, saddle point, cannot be
determined)
The critical point with the next smallest x-coordinate is
( , 3) Classification: (local minimum, local maximum, saddle point, cannot be
determined)
The critical point with the next smallest x-coordinate is
(18, 0) Classification: (local minimum, local maximum, saddle point, cannot be
determined)
The critical point with the next smallest x-coordinate is
(18, ) Classification: (local minimum, local maximum, saddle point, cannot be
determined)
