Let $f: \mathbb{C} \to \mathbb{C} $ such that $f(x+iy)=x^3+3xy^2+i(y^3+3x^2y)$

Question

Let $f: \mathbb{C} \to \mathbb{C} $ such that

$$f(x+iy)=x^3+3xy^2+i(y^3+3x^2y)$$

let $f'(z)$ be the derivative of $f(z)$ ,Then which of the following is true

1)$f'(1+i)$ exists and $|f'(1+i)=3\sqrt5$

$2).f $ is analytic at origin

$3).f$ is not differbtiable at $i$

$4).f$ is differentiable at $1$

I tried to solve the question by using CR equation , $$u_x=3x^2+3y^2$$ $$u_y=6xy$$ $$v_x=6xy$$ $$v_y=3y^2+3x^2$$

form above i get that this is not analytic because $u_y \neq -v_x$ thus option $2$ and $4$ are not true further i am not getting how to check for $1$ and $3$ option

Please help

Please help

Answer 1

The solutions of the Cauchy-Riemann equations are the numbers $x+yi$ such that $x=0$ or $y=0$. Therefore, the correct options is the fourth one. The other options are wrong.