The Function:
The given function is f(x, y) = 2x^4 - 3x^2y + y^2.

Finding Extrema:
To determine whether the function has a maximum or minimum at a particular point, we need to find the critical points. Critical points occur where the derivative of the function is equal to zero or undefined.

Finding the Partial Derivatives:
To find the critical points, we need to take the partial derivatives of f(x, y) with respect to x and y.

Taking the partial derivative of f(x, y) with respect to x:
∂f/∂x = 8x^3 - 6xy

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

Solving for Critical Points:
Setting both partial derivatives equal to zero, we can solve for the critical points:

8x^3 - 6xy = 0 ...(1)
-3x^2 + 2y = 0 ...(2)

Solving equation (1) for x, we get:
8x^3 = 6xy
x^2 = 3y/4
x = ±√(3y/4)

Substituting the value of x in equation (2), we have:
-3(√(3y/4))^2 + 2y = 0
-3(3y/4) + 2y = 0
-9y/4 + 2y = 0
-9y + 8y = 0
-y = 0
y = 0

Substituting y = 0 into equation (2), we get:
-3x^2 + 2(0) = 0
-3x^2 = 0
x^2 = 0
x = 0

Therefore, the critical point is (0, 0).

Classifying the Critical Point:
To classify whether the critical point (0, 0) is a maximum, minimum, or neither, we need to use the second derivative test.

Taking the second partial derivatives:

∂^2f/∂x^2 = 24x^2 - 6y
∂^2f/∂y^2 = 2

Calculating the mixed partial derivative:

∂^2f/∂x∂y = -6x

Using the Second Derivative Test:
Plugging the critical point (0, 0) into the second partial derivatives, we have:
∂^2f/∂x^2 = 24(0)^2 - 6(0) = 0
∂^2f/∂y^2 = 2
∂^2f/∂x∂y = -6(0) = 0

The determinant of the Hessian matrix is:
D = (∂^2f/∂x^2)(∂^2f/∂y^2) - (∂^2f/∂x∂y)^2 = (0)(2) - (0)^2 = 0

Since the