Let $f$ be the function defined on $R^2$ by $f(x, y) = x^5+y^3-x^2+2x-3y.$
(a) Show that $f$ is of class $C^\infty$ on $R^2$ and compute its gradient at a point $(x, y)\in R^2.$
(b) Show that, in a neighbourhood of $(0, 0)$ the equation $x^5+y^3-x^2+2x-3y=0$ defines $y$ as a $C^\infty$ function $y=\phi(x)$ where $\phi(0)=0.$
(c) Write the Taylor formula of degree 3 for $\phi$ about the origin $(0, 0).$
I'm interested in c so let's assume i have done a and b. I rewrite the equation to $$3\phi(x)=x^5+\phi(x)^3-x^2+2x=\phi(x)^3-x^2+2x+O(x^5)$$ i then find the taylor polynomial and see that $$\phi(x)=f(0)+f_x(0)x+O(x^2)=0-(2\cdot0+2)x+O(x^2)=2x+O(x^2)$$ i now use the implicit function theorem $$\frac{\partial f}{\partial x}(0,0)=-3$$ and divide with the factor $-(-3)=3$ $$\phi(x)^3=({2x\over3})^3+{O(x^2)^3\over3}={8x\over 27}+{O(x^2)^3\over3}$$ So i get $$\phi(x)={-x^2+2x+{8x\over27}+O(x^5)\over 3}$$
Is this correct?