Find all the local maxima local minima and saddle points of

Find all the local maxima, local minima, and saddle points of the function f(x, y) = x^3 + y^3 + 3x^2 - 6y^2 - 2 Select the correct choice below and. if necessary, fill in the answer boxes to complete your choice A local maximum occurs at (Type an ordered pair Use a comma to separate answers as needed) The local maximum value(s) is/are (Type an exact answer Use a comma to separate answers as needed) There are no local maxima

Solution

f_x = 3x² + 6x f_xx = 6x + 6 = 6(x + 1)

f_y = 3y² - 12y f_yy = 6y - 12 = 6(y - 2)

f_xy = f_yx = 0

Equating f_x and f_y equals to zero

f_x = 0 3x² + 6x = 0 x = 0 or -2

f_y = 0 3y² - 12y = 0 y = 0 or 4

Critical Points : ( 0 , 0 ) ; ( 0 , 4 ) ; ( -2 , 0 ) ; ( -2 , 4 )

Calculating

f_xx . f_yy - f_xy² = 36 ( x + 1 ) ( y - 2 )

At ( 0 , 0 )

f_xx . f_yy - f_xy² < 0 Saddle Point

At ( 0 , 4 )

f_xx . f_yy - f_xy² > 0 Maxima/Minima

f_xx > 0 and f_yy > 0 Minima

At ( -2 , 0 )

f_xx . f_yy - f_xy² > 0 Maxima/Minima

f_xx < 0 and f_yy < 0 Maxima

At ( -2 , 4 )

f_xx . f_yy - f_xy² < 0 Saddle Point