To determine the nature of the critical points of the function f(x, y) = x^3 - y^3 - 3x + 3, we need to find the partial derivatives with respect to x and y and set them equal to zero.

∂f/∂x = 3x^2 - 3 = 0
∂f/∂y = -3y^2 = 0

From the first equation, we have 3x^2 - 3 = 0, which simplifies to x^2 - 1 = 0. Solving for x, we find x = ±1.

From the second equation, we have -3y^2 = 0, which implies y = 0.

Therefore, the critical points are (1, 0) and (-1, 0).

To determine the nature of these critical points, we need to evaluate the second partial derivatives.

∂²f/∂x² = 6x
∂²f/∂y² = -6y
∂²f/∂x∂y = 0

Evaluating the second partial derivatives at the critical points (1, 0) and (-1, 0):

At (1, 0):
∂²f/∂x² = 6(1) = 6 > 0
∂²f/∂y² = -6(0) = 0
∂²f/∂x∂y = 0

At (-1, 0):
∂²f/∂x² = 6(-1) = -6 < 0
∂²f/∂y² = -6(0) = 0
∂²f/∂x∂y = 0

Using the second partial derivative test, we can conclude:

At (1, 0): The second partial derivative test is inconclusive. Further analysis is needed.

At (-1, 0): The second partial derivative test indicates a saddle point.

Therefore, the critical point (1, 0) requires further analysis to determine its nature.