Explanation:

The given function is f(x, y) = x^2 - 2xyy^2 - x^3 + y^3 - x^5

To determine whether the function has a maximum or minimum value at the origin (0, 0), we need to analyze the behavior of the function in the neighborhood of the origin.

First-Order Partial Derivatives:
To find the critical points, we take the first-order partial derivatives of f(x, y) with respect to x and y and set them equal to zero.

∂f/∂x = 2x - 2yy^2 - 3x^2 - 5x^4 (Partial derivative with respect to x)
∂f/∂y = -2xy - 4yy^2 + 3y^2 (Partial derivative with respect to y)

Setting both partial derivatives equal to zero, we have:

2x - 2yy^2 - 3x^2 - 5x^4 = 0 (Equation 1)
-2xy - 4yy^2 + 3y^2 = 0 (Equation 2)

Second-Order Partial Derivatives:
To determine the nature of the critical points, we need to find the second-order partial derivatives of f(x, y).

∂^2f/∂x^2 = 2 - 6x - 20x^3 (Second partial derivative with respect to x)
∂^2f/∂y^2 = -2x - 8yy + 6y (Second partial derivative with respect to y)
∂^2f/∂x∂y = -2y - 8yy + 6y (Mixed partial derivative)

Hessian Matrix:
The Hessian matrix is a matrix of second-order partial derivatives:

H = | ∂^2f/∂x^2 ∂^2f/∂x∂y |
| ∂^2f/∂x∂y ∂^2f/∂y^2 |

Evaluating the Hessian matrix at the origin (0, 0), we have:

H(0, 0) = | 2 0 |
| 0 0 |

Determinant of the Hessian Matrix:
The determinant of the Hessian matrix can be calculated as follows:

|H(0, 0)| = |2 0 |
|0 0 | = 2(0) - 0(0) = 0

Conclusions:
1. If the determinant of the Hessian matrix is positive, then the critical point is a local minimum.
2. If the determinant of the Hessian matrix is negative, then the critical point is a local maximum.
3. If the determinant of the Hessian matrix is zero, then the test is inconclusive.

Since the determinant of the Hessian matrix at the origin is zero, the test is inconclusive. Therefore, the function f(x, y) = x^2 - 2xyy^2 - x^3 + y^3 - x^5 has neither a maximum nor a minimum value at