Find the critical points for the function The critical point

Find the critical points for the function

The critical points found are correct, but I am not sure what the classification is.

(1 pt) Find the critical points for the function and classify each as a local maximum, local minimum, saddle point, or none of these. critical points: (0,-1),(0,1),(8,-1),(8,1) give your points as a comma separated list of (x,y) coordinates.) classifications: give your answers in a comma separated list, specifying maximum, minimum, saddle point, or ove your points as a comseaeds o kay)/coordinates.) none for each, in the same order as you entered your critical points)

Solution

f_x = 3 x^2 - 24 x

f_xx = 6x -24

f_y = 3y^2 - 3

f_yy = 6y

now f_x = 0 x = 0 , 8

f_y = 0 y = 1 ,-1

so (0,1),(0,-1),(8,1),(8,-1) are critical points.

now f_xx ( 0) < 0 ,  f_xx (8) > 0

f_yy (-1) < 0 ,f_yy(1) > 0

hence local maximum {where fxx < 0  & f_yy < 0 } = (0,-1)

hence local minimum {where  fxx > 0  & f_yy >0 } = (8,1)

rest two points are saddle points