Explanation:

To determine which of the given functions cannot be the real part of a complex analytic function, we need to analyze the Cauchy-Riemann equations.

The Cauchy-Riemann equations state that if f(z) = u(x, y) + iv(x, y) is a complex analytic function, then the partial derivatives of u and v with respect to x and y must satisfy the following conditions:

1. ∂u/∂x = ∂v/∂y
2. ∂u/∂y = -∂v/∂x

We will examine each option one by one to see if the given function satisfies the Cauchy-Riemann equations.

Option A: 3x^2y - y - y^3
To check if this function satisfies the Cauchy-Riemann equations, we need to calculate the partial derivatives:

∂u/∂x = 6xy
∂u/∂y = 3x^2 - 1 - 3y^2

∂v/∂x = 0 (since there is no variable 'v' in the function)
∂v/∂y = 0 (since there is no variable 'v' in the function)

Comparing the partial derivatives, we can see that ∂u/∂x ≠ ∂v/∂y and ∂u/∂y ≠ -∂v/∂x. Therefore, this function does not satisfy the Cauchy-Riemann equations and cannot be the real part of a complex analytic function.

Option B: x^2y
To check if this function satisfies the Cauchy-Riemann equations, we need to calculate the partial derivatives:

∂u/∂x = 2xy
∂u/∂y = x^2

∂v/∂x = 0 (since there is no variable 'v' in the function)
∂v/∂y = 0 (since there is no variable 'v' in the function)

Comparing the partial derivatives, we can see that ∂u/∂x ≠ ∂v/∂y and ∂u/∂y ≠ -∂v/∂x. Therefore, this function does not satisfy the Cauchy-Riemann equations and cannot be the real part of a complex analytic function.

Option C: x^3 - 3xy^2
To check if this function satisfies the Cauchy-Riemann equations, we need to calculate the partial derivatives:

∂u/∂x = 3x^2 - 3y^2
∂u/∂y = -6xy

∂v/∂x = 0 (since there is no variable 'v' in the function)
∂v/∂y = 0 (since there is no variable 'v' in the function)

Comparing the partial derivatives, we can see that ∂u/∂x ≠ ∂v/∂y and ∂u/∂y ≠ -∂v/∂x. Therefore, this function does not satisfy the Cauchy-Riemann equations and cannot be the real