# Tagged Questions

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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### Find all local extremums of $f(x)=x^{2}e^{-x}$ and decide if these are global extremums

As all my other questions, this one isn't homework (it's preparation for an exam). I'd like to know if I did everything correctly. In my previous task, I had a mistake in the first derivation. But ...
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### Find all local extremums of $f(x)=\frac{x}{x^{2}+x+1}$ and decide if these are global extremums

Did I do everything correctly? Find all local extremums of the following function and decide if these are global extremums (i.e. maxima or rather minima of the function on its entire domain) ...
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### Convergence of a integral for every curve in the sphere

Let $S$ be the unit open sphere in $\mathbb{R}^3$: $x^2+y^2+z^2< 1$ and $\partial S$ its border $x^2+y^2+z^2= 1$. Let $f:S\cup \partial S\rightarrow \mathbb{R}$ be a continuous function which is ...
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### Is the mathematical syntax correct here (Taylor-polynomial)?

Say I'm supposed to create the $2^{nd}$ degree Taylor-polynomial of $f(x) = \cos x$ at $x_{0} = 0$ I'd like to know if the syntax is correct, how I solved this little task. We have defined the ...
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### Does there exist any non-zero polynomial $f:\mathbb C \to \mathbb C$ such that $f(x+2)-2f(x+1)=f(x) , \forall x \in \mathbb C$ ?

Does there exist any non-zero polynomial $f:\mathbb C \to \mathbb C$ such that $f(x+2)-2f(x+1)=f(x) , \forall x \in \mathbb C$ ?
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### Cauchy Product Radius of Convergence

How do we see that the radius of convergence (RCV) of the Cauchy product is at least the minimum of the two respective RCV's? For instance suppose $f(x)=\sum_{i=0}^\infty a_i(x-x_0)^i$ with RCV $R_1$....
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### statistical comparison, 3 groups, multiple columns

I am using R for some statistical analysis. I have a dataset listing number of deaths by eu regions. the dataset is annual and is for 2000-2008. I divided this data into 4 subgroups according to ...
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### If $m^*([-n,n] \cap E) + m^*([-n,n] \setminus E) = 2n$ for all $n$, then $E$ is Lebesgue measurable

Let $E \subset \Bbb R$ and let $m^*$ denote the Lebesgue outer measure on $\Bbb R$. Show that if for all $n \in \Bbb N$, $m^*([-n,n] \cap E) + m^*([-n,n] \setminus E) = 2n$, then $E$ is Lebesgue ...
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### The Jeep Problem and Nash's Friends

The classical jeep problem is the following. A jeep can carry a maximum load of fuel of 1 gallon, and it travels $l$ miles with $l$ gallons of fuel. The jeep moves along a straight line, and is ...
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### Dual of the Banach space of $k$-times continuously differentiable functions.

Let $C^k([0,1])$ denote the Banach space of $k$-times continuously differentiable functions $f:[0,1]\to \mathbb R$ with norm $$\|f\|_{C^k}:=\max_{i=0,\dots,k}\sup_{x\in [0,1]}|f^{i}(x)|.$$ I'm trying ...
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### Diagram of a multivaribale function

I have to draw the diagram of the function: $$(x^2+y^2)^{\frac{3}{2}}=x^2-y^2$$ I transformed it with polar coordinates to: $$r=\cos^2(\varphi)-\sin^2(\varphi)$$ with $r \ge 0$ and $\cos^2 \ge sin^2$....
### $\text{ Proving }\; A \subseteq \Bbb R \text{ A is bounded above} \Rightarrow A^c \text{ is not?}$
Prove: Let $A \subseteq \Bbb R$. Prove that if $A$ is bounded above, then $A^c$, the complement of $A$ is not bounded above. $A^c =$ those element of the universe that are not in A. $\Bbb R =$ ...