Thread: Prove solution exists

1. Prove solution exists

Let $f: \mathbb{R}^{k+n} \rightarrow \mathbb{R}^n$ be of class $C^1$; suppose that f(a) = 0 and that Df(a) has rank n. Show that if c is a point of $\mathbb{R}^n$ sufficiently close to 0, then the equation f(x) = c has a solution.

I'm pretty sure that I have to use the implicit function theorem here, but I'm not sure how to proceed. Any ideas?

2. Originally Posted by JG89
Let $f: \mathbb{R}^{k+n} \rightarrow \mathbb{R}^n$ be of class $C^1$; suppose that f(a) = 0 and that Df(a) has rank n. Show that if c is a point of $\mathbb{R}^n$ sufficiently close to 0, then the equation f(x) = c has a solution.

I'm pretty sure that I have to use the implicit function theorem here, but I'm not sure how to proceed. Any ideas?

As you can see in "statement of the theorem" in http://en.wikipedia.org/wiki/Implicit_function_theorem" , or

perhaps a little clearer in Implicit Function Theorem -- from Wolfram MathWorld ,

this is almost the exact wording of the implicit function theorem, so I don't understand what is it

that you cannot understand here?

Perhaps it is just a matter of writing $a=(x_1,...,x_n,x_{n+1},...,x_{n+m})\in\mathbb{R}^{ n+m}$ so that it'll look clearer...?

Tonio

3. Let $a = (a_1, a_2)$ The theorem says that since f(a) = 0 and $\frac{\partial f}{\partial y} (a)$ has rank n (this follows from Df(a) having rank n), then there is a neighborhood B of $a_1$ in $\mathbb{R}^k$ and a $C^r$ map $g: B \rightarrow \mathbb{R}^n$ such that $g(a_1) = a_2$ and f(x, g(x)) = 0 for all x in B.

How does this show that if c is a point very close to 0 ( but not quite 0), that there is a solution to f(x) = c?