Let $f \colon \Bbb C \to \Bbb C$ be a complex valued function given by $f(z)=u(x,y)+iv(x,y).$ I am stuck on the following question :  
 
MY ATTEMPT:
By Cauchy Riemann equation ,we have $u_x=v_y,u_y=-v_x.$ Now $v(x,y)=3xy^2 \implies v_x=3y^2 \implies -u_y=3y^2 \implies u=-y^3+ \phi(x) $. Now,I am not sure which way to go? Can someone give some explanation about which way to go in order to pick the correct option?

If this function is holomorphic, than $u(x,y)$ and $v(x,y)$ must be harmonically adjoint. That means, that $v(x,y)$ must satisfy 
$$\Delta v(x,y)=0; $$
We found the values of $\frac{\partial^2 v}{\partial x^2}$, $\frac{\partial^2 v}{\partial y^2}$
$$\frac{\partial^2 v}{\partial x^2}=0; \frac{\partial^2 v}{\partial y^2}=6x $$
So, the next equality must be true for all $(x,y)\in\mathbb{C}$
$$6x=0 $$
But this is true only for $x=0.$ So, this function can not be holomorphic for any $u(x,y)$
