Consider a function $f : R^2 \rightarrow R$ given by $f(x,y)= x^2 - y^3.$ I need to check whether the hypothesis of Implicit function theorem hold for $f$ at the point $(0,0)$.
For this, I first calculated the matrix $M(x,y)$ formed by differentiating this function with respect to $y$. $M(x,y) = -3y^2$. It's clearly non-invertible at $(0,0)$. I am stuck here as non-invertibility gives me an inconclusive test. How do I show that hypothesis of Implicit function theorem holds here or not?
Your calculations are correct. So, you have verified that the hypotheses of the Implicit Function Theorem are not satisfied by $f$ at the point $(0,0)$.
Additionally, note that at any point $(x,y) \neq (0,0)$, either $D_1 f(x,y) \neq 0$ or $D_2 f(x,y) \neq 0$. So, the hypotheses of the Implicit Function Theorem will be satisfied by $f$ at any point $(x,y)$ such that $f(x,y) = 0$ except $(0,0)$.