Find the critical points of the equation

The critical points I found in increasing lexicographic order are

(0.0) (0,1/9),(1/21,1/27) and (1/7, 0)

at (0,0) is it undefined

at (0/1/9) and (1/7,0) you have saddle points

at (1/21,1/2) you have a local max.

Is this correct?

Thank you