Is it analytic ? $f(z)=z^2\bar{z}$ Please find the question and its solution.



My answer gone wrong dont know why, I think the answer marked in the question is wrong.
My try:
I solved for CR equations and replaced x=y=0 and found CR equations satisfied. But not sure why in the solution it says CR equations doesn't satisfy.
Please let me know what should be the correct answer ?
You are right: since $u_x(0,0)=u_y(0,0)=v_x(0,0)=v_y(0,0)=0$, $(0,0)$ is a solution of the Cauchy-Riemann equations (and, since the partial derivatives are continuous at $(0,0)$,  $f$ differentiable at $0$).
Apart from the continuity of all partial derivatives $u_x, v_x, u_y, v_y$ along with $u$ and $v$ everywhere, note that $u_x=v_y \forall x,y$
But $u_y=-v_x$ holds for $xy=0\implies x=0$ or $y=0$ i.e. on coordinate axes. So

CR-equations hold at $z=0$ and the test says that $f(z)$ is differentiable at $z=0$ (not analytic though).


I would say that the question is poorly formatted as all the options stand incorrect.