Question

23 Find the local mportant aspects of the function. (Enter your answers as a and graph the and list. If an answer does not exist, enter DNE saddle point(s) (0,1),(0,-10x and important aspects of the function. (Enter your answers as a -59 xn(3,0,-27) x

Answer 1

f(x, y) = x^3 - 9x + 3xy^2 partial differential f/partial differential x = 3x^2 - 3 + 3y^2, partial differential^2 f/partial differential x partial differential y = 6y, partial differential^2 f/partial differential x^2 = 6x partial differential f/partial differential y = 6xy, partial differential^2 f/partial differential y^2 = 6x 3x^2 - 3 + 3y^2 x^2 + y^2 = 1 6xy = 0 xy = 0 D = f_xx f_yy - (f_xy)^2 = 36xy - 36x^2 = 36x (y - x) x^2 + y^2 + 2xy = 1 (x + y)^2 = + 1 x + y = plusminus 1 x = -y plusminus 1 We only for saddle p: - because local minimum and maximum pt are given and constant For, the pt (1, 0), and (-1, 0) then D = -36 < 0 Therefore (1, 0) and (-1, 0) are saddle pt Therefore f (1, 0) = 1 - 3 = -2, f(-1, 0) = -1 + 3 = 2 f(x, y) = x^3