The function given is $f(z) = (x^2 + y) + i(y^2 - x)$, where $z = x + iy$. To determine where $f(z)$ is analytic, we need to check whether the Cauchy-Riemann equations hold. The Cauchy-Riemann equations are:

$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

Here, $u(x, y) = x^2 + y$ and $v(x, y) = y^2 - x$.

Let's compute the partial derivatives:

• $\frac{\partial u}{\partial x} = 2x$
• $\frac{\partial u}{\partial y} = 1$
• $\frac{\partial v}{\partial x} = -1$
• $\frac{\partial v}{\partial y} = 2y$

Now, substitute these into the Cauchy-Riemann equations:

1. $\frac{\partial u}{\partial x} = 2x$ and $\frac{\partial v}{\partial y} = 2y$. For the first equation to hold, $2x = 2y$, which implies $x = y$.
2. $\frac{\partial u}{\partial y} = 1$ and $-\frac{\partial v}{\partial x} = 1$. For the second equation to hold, we have $1 = 1$, which is always true.

Thus, the Cauchy-Riemann equations are satisfied when $x = y$, which corresponds to the line $z = x + iy$ where $x = y$. Therefore, the function $f(z)$ is only analytic along the line where $x = y$.

Among the choices, the most appropriate option is "f is not analytic at any point", because the function fails to satisfy the Cauchy-Riemann equations except on a line, not at any specific point.

Would you like further clarification or details? Here are some related questions to expand your understanding:

1. What are the Cauchy-Riemann equations, and how are they used to test analyticity?
2. Can a function be analytic along a curve but not at isolated points?
3. What is the significance of a function being analytic at every point in a domain?
4. How do harmonic functions relate to analytic functions in complex analysis?
5. What are the implications of a function not being analytic in the context of complex functions?

Tip: When testing a function for analyticity, always start by checking if the Cauchy-Riemann equations hold. This gives a quick way to determine whether the function is differentiable at a point in the complex plane.