Implicit function theorem - C\infty

[CnstMot]

Be \(\displaystyle f: U\rightarrow \mathbb{R}\) continious, such that \(\displaystyle (x^2+y^{4})f+f^{3}=1\) in \(\displaystyle U\subset \mathbb{R}^{n}\) open. Show that \(\displaystyle f\) is \(\displaystyle C^{\infty}\).

 

[daon2]

Be \(\displaystyle f: U\rightarrow \mathbb{R}\) continious, such that \(\displaystyle (x^2+y^{4})f+f^{3}=1\) in \(\displaystyle U\subset \mathbb{R}^{n}\) open. Show that \(\displaystyle f\) is \(\displaystyle C^{\infty}\).

The first step would be showing that \(\displaystyle f\) is at least \(\displaystyle C^1\). Then you can find \(\displaystyle Df\) in terms of \(\displaystyle f\) showing it is of class \(\displaystyle C^2\), and this implies \(\displaystyle C^{\infty}\).

edit: I see how to use the implicit function theorem. You want to find a function \(\displaystyle G:\mathbb{R}^{2+n}\to\mathbb{R}^n\), of the form \(\displaystyle G(t,z)\) where \(\displaystyle z=(z_1,...,z_n)\in\mathbb{R}^n, t=(x,y)\in \mathbb{R}^2\). Of course you will want to use your function \(\displaystyle f\) in the definition of \(\displaystyle G\).

Then \(\displaystyle DG = \left[\partial G/\partial(x,y)\,\, \partial G/\partial z\right]\). You will use the implicit function theorem to show that \(\displaystyle z=h(x,y)\) noting that \(\displaystyle (x^2+y^4)f(z)+f(3)^3-1 = 0\) for all values of x,y,z.