List the critical points of the functionfxy x 95 y53 for

List the critical points of the function:f(x,y) = (x - 9)^5 * (y+5)^3, for which the Second Partial Test fails.

Solution

f(x,y) = (x-9)⁵(y+5)³

Find the critical points by solving the simultaneous equations

f_x(x, y) = 0; f_y(x, y) = 0.

f_x = 5(x-9)⁴(y+5)³

f_y = 3(x-9)⁵(y+5)²

Both partial derivatives are zero if x = 9 or y = -5

f_xx = 20(x-9)³(y+5)³

f_yy = 6(x-9)⁵(y+5)

f_xy = 15(x-9)⁴(y+5)²

Also D = f_xxf_yy - f_xy² = 0 for x = 9 or y = -5

Hence the second partials test fails for for every point on the line x = 9 and y = -5