# Tagged Questions

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### The inverse of a Moment generating function

The moment generating function of $X$ is $M_X(t) = \mathbb{E}[e^{tX}] = \int e^{tu}f_X(u)du$ where t is a complex variable and $f_X$ is the density of X. The cumulant generating funtion of $X$ is ...
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### A formula for n-derivative of the inverse of a function?

Let $y=f^{-1}(x)$. As we know: \begin{align} \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{1}{{f}'(y)} \end{align} Thereof we have: \begin{align} \frac{\mathrm{d^2} y}{\mathrm{d} ...
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### 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 ...
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### Proof with functions and inverse - Spivak

How does he know that $f^{-1}$ is one-one? Doesn't he have to prove that? Or is he applying his first theorem in the chapter to $f$? That is $f$ is a function if and only if $f^{-1}$ is ...
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### Intermediate Value Theorem, least upper bounds, Spivak

I have only taken an excerpt from the book from Spivak 3rd edition page 220 in his "Inverse Function" chapter. At the end of the 3rd paragraph, he says that Then $f$ takes on some value ...