find the critical points of the function ( f(x,y)=x^{3}-6x^{2}-2y^{3}+3y^{2}+4 ). list your answers as…

Explanation:

Step1: Find the partial derivatives

The partial derivative with respect to (x): (f_x = 3x^2 - 12x) The partial derivative with respect to (y): (f_y=-6y^2 + 6y)

Step2: Set the partial derivatives equal to zero

For (f_x = 0): (3x^2-12x = 0) Factor out (3x): (3x(x - 4)=0) So (x = 0) or (x = 4)

For (f_y=0): (-6y^2 + 6y=0) Factor out (-6y): (-6y(y - 1)=0) So (y = 0) or (y = 1)

Step3: Combine the solutions

When (x = 0), (y = 0) gives the point ((0,0)); (y = 1) gives the point ((0,1)) When (x = 4), (y = 0) gives the point ((4,0)); (y = 1) gives the point ((4,1))

Answer:

((0,0),(0,1),(4,0),(4,1))