Question

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

f(x, y) = x³ − 27xy + 27y³

Answer 1

critical point ( maximum ,minimum or saddle point ) of a function occur accordingly

     f_x ( partial differentiation with respect to x ) = 0

and fy ( partial differentiation with respect to y ) = 0

then calculate D = f_xx .f_yy - (f_xy)² at the point where f_x = 0 and f_y=0

where f_xx => partially differentiate f_x with respect to x

          f_yy => partially differentiate f_y with respect to y

          f_xy => partially differentiate f_x with respect to y

If D>0 and f_xx > 0 => relative minimum

If D>0 and f_xx < 0 => relative maximum

If D<0 => saddle point

If D = 0 then no conclusion can be drawn