Answer to Question #224074 in Calculus for Unknown346307

Question

Evaluate and classify the critical point of the function f(x, y) = xy

Answer 1

Given f(x,y) = xy

then f_x = y and f_y = x

for critical points, f_x = 0 and f_y = 0

$\implies$ x = 0 and y = 0

therefore the only critical point is (0,0)

we know that f_x = y and f_y = x

$\implies$ f_xx = 0, f_yy = 0 and f_xy = 1

Hence, f_xx$\cdot$ f_yy - ( f_xy )² = 0 - 1 = -1 < 0

(0,0) is a saddle point.