- #1

Calabi

- 140

- 2

## Homework Statement

Let be ##f \in C^{1}(\mathbb{R}^{2}, \mathbb{R})## and ##(x_{0} , y_{0}) \in \mathbb{R}^{2} / f(x_{0}, y_{0}) = 0##. I suppose ##|\frac{\partial f}{\partial y}| \ge |\frac{\partial f}{\partial x}|(2)##.

I'm looking for ##\phi \in C^{1}(\mathbb{R}, \mathbb{R}) / \forall x \in \mathbb{R} f(x, \phi(x)) = 0## and ##\phi(x_{0}) = y_{0}##.

## Homework Equations

##(x_{0} , y_{0}) \in \mathbb{R}^{2} / f(x_{0}, y_{0}) = 0##

##|\frac{\partial f}{\partial y}| \ge |\frac{\partial f}{\partial x}|##

## The Attempt at a Solution

If a such solution exists we necssarly have ##\forall x \in \mathbb{R}, \phi'(x) \frac{\partial f}{\partial y}(x, \phi(x)) + \frac{\partial f}{\partial x}(x, \phi(x)) = 0 (1)##.

We can stated ##x \rightarrow (x, \phi(x))## is injective so if we look its image we perhaps make a variable changement in (1). But I don't think it's usefull.

And I don't see how the majoration (2) is usefull.