2
votes
0answers
31 views

checking whether functions satisfy Inverse Function Theorem.

I've my exam tomorrow and this question is expected to come but donot know how to solve... Here's the INVERSE FUNCTION THEOREM stated in my notes: It says: Let $E\subseteq \mathbb R^n$ be open ...
3
votes
1answer
46 views

Closed form of the inverse of a function

Does anyone know what the analytic form of the inverse of $f(x)=e^x+x$? Thanks in advance
0
votes
0answers
32 views

Local inversion theorem (théorème d'inversion local)

I don't understand how to use the local inversion theorem to prove that a nondegerate critical point of a function $f\in C^2(U,\mathbb{R})$ is isolated Thank you.
2
votes
2answers
50 views

prove function is surjective /analysis proofs!!

Suppose $f:(a,b)\longrightarrow\mathbb R$, differentiable, where $(a,b)\subseteq\mathbb R$ is an open interval. Assume that $f'(x)$ is not $=0$. Show that there is an open interval ...
4
votes
2answers
239 views

Taylor series of the inverse of $x^4+x$

I would like to expand the inverse function of $$g(x) := x^4+x $$ in a taylor series at the point x = 0. I calculated the first and second derivate at x = 0 with the rule of the derivation of an ...
1
vote
1answer
25 views

Derivation of inner variations

In Giaquinta's and Hildebrandt's 1996, "Calculus of Variations 1", pages 147-148, they develop the definition of inner variations. They first fix $\lambda\in ...
1
vote
3answers
152 views

Do continuous mappings always have an inverse?

A theorem of general topology states that: A mapping $f$ from $X$ to $Y$ is continuous if and only if the inverse image of any open set in $Y$ is open in $X$. Does this mean that continuous ...
0
votes
1answer
43 views

The differentiability class of the inverse function

Here's the final part of a proof (from Marden's Elementary Classical Analysis) of the inverse function theorem, where we have been given that $f$ is of class $C^p$: Could someone please explain the ...
2
votes
2answers
40 views

How to prove that $f$ is $1-1$ from $E$ on $\{ (s,t) : s> 2\sqrt{t} >0\}$

Question: Let $E=\{(x,y): 0<y<x \}$ set $f(x,y)=(x+y, xy)$ for $(x,y)\in E$ a) How to prove that $f$ is $1-1$ from $E$ on $\{ (s,t) : s> 2\sqrt{t} >0\}$ And how to find formula for ...
3
votes
1answer
71 views

Why $ g(p) = 0.5 p^{-0.2} + 0.5 p^{-0.5} $ has a well-defined inverse that is continuous and strictly decreasing.

A book that I am reading claims the following about the function $ g(p) = 0.5 p^{-0.2} + 0.5 p^{-0.5} $ (which is a demand function): Formal arguments based on the Intermediate Value Theorem and ...
0
votes
1answer
58 views

Relationship between real inverses of analytic functions

Take some analytic function, $f(x)$, that goes from $-\infty$ to $\infty$, with a finite number of points such that $\frac{df}{dx}=0$. You can divide the y axis into intervals, where the boundary ...
2
votes
2answers
197 views

$f(x,y)$ in polar coordinates

So, I have $ f(x,y) = (x^2-y^2, 2xy) $, which is a local $\mathcal C^1$ isomorphism in $\mathbb R^2 \setminus \{(0,0)\}$. I have to write this function in polar coordinates: $$f(x,y) = ...