Given Function:
The given function is:
f(x,y) = x^3y^3 - 63xy + 12xy

Stationary Points:
To find the stationary points of the function, we need to find the values of x and y where the partial derivatives of the function with respect to x and y are equal to zero.

Taking the partial derivative of f(x,y) with respect to x:
∂f/∂x = 3x^2y^3 - 63y + 12y

Taking the partial derivative of f(x,y) with respect to y:
∂f/∂y = 3x^3y^2 - 63x + 12x

Setting both partial derivatives equal to zero and solving for x and y, we get:
3x^2y^3 - 63y + 12y = 0
3x^3y^2 - 63x + 12x = 0

The Function has Three Stationary Points:
To determine the number of stationary points, we need to solve the system of equations formed by setting the partial derivatives equal to zero.

By solving the system of equations, we can find the values of x and y where the function has stationary points. If the system has three distinct solutions, then the function has three stationary points.

The Function is Minimum at (-7, -7):
To determine whether the function is a minimum or maximum at a specific point, we need to consider the second-order partial derivatives of the function.

Calculating the second-order partial derivatives of f(x,y):
∂^2f/∂x^2 = 6xy^3
∂^2f/∂y^2 = 6x^3y
∂^2f/∂x∂y = 9x^2y^2 - 63
∂^2f/∂y∂x = 9x^2y^2 - 63

To determine the nature of the critical point, we can use the second-order partial derivatives test. However, in this case, we can see that the second-order partial derivatives are not needed to determine the nature of the point.

The Function is Maximum at (3, 3):
Similarly, to determine whether the function is a minimum or maximum at a specific point, we need to consider the second-order partial derivatives of the function.

Calculating the second-order partial derivatives of f(x,y):
∂^2f/∂x^2 = 6xy^3
∂^2f/∂y^2 = 6x^3y
∂^2f/∂x∂y = 9x^2y^2 - 63
∂^2f/∂y∂x = 9x^2y^2 - 63

To determine the nature of the critical point, we can use the second-order partial derivatives test. However, in this case, we can see that the second-order partial derivatives are not needed to determine the nature of the point.

The Function has Neither Minimum nor Maximum at (5, -1):
To determine whether the function has a minimum or maximum at a specific point, we need to consider the