Cubic-over-quadratic partials at origin

Let $f:\mathbb{R}^{2}\to\mathbb{R}$ be a function defined by $$f(x,y)=\begin{cases}\dfrac{y^{3}}{x^{2}+y^{2}},&(x,y)\neq(0,0)\\0,&(x,y)=(0,0)\end{cases}.$$ Let $f_{x}(x,y)$ and $f_{y}(x,y)$ denote the first order partial derivatives of $f$ with respect to $x$ and $y$, respectively, at the point $(x,y)$. Then, which of the following statements is FALSE?