# Tagged Questions

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### How to show $f(x,y) \leq \theta f(x,y) + (1-\theta)f(x,y)$ for $\theta \in [0,1]$?

Let $\theta \in [0, 1]$. Let $f(x,y)$ be a function. Is there a way I could prove that $f(x,y) \leq \theta f(x,y) + (1-\theta)f(x,y)$? I have tried to start with $f(x,y) = 2f(x,y) - f(x,y)$ or ...
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### Establish the absolute maximum of a function

We have this function:$$f(x)=\begin{cases} \sin(x) \cdot\ln(\sin2x), & \mbox{if }0<x<\pi/2 \\ 0, & \mbox{if }x=0,\mbox{or }x=\pi/2 \end{cases}$$ So, how to prove that it decreases and ...
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### Fractional derivative of exponential function

With the $n$th order derivative ($n$ as a positive integer) of $e^{ax}$ given by $$D^{n}e^{ax}=a^ne^{ax},$$ is the generalized (or fractional) derivative the same? Does it apply for any arbitrary ...
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### There cannot exist a rational function $f: \mathbb{R} \to \mathbb{R}$ injective, not surjective

I was looking for a rational function $f: \mathbb{R} \to \mathbb{R}$ that looks like $\arctan$, in that it is injective not surjective well-defined on all $x\in \mathbb{R}$ (no vertical ...
### A function satisfying $f(\frac1{x+1})\cdot x=f(x)-1$ and $f(1)=1$?
$f:[0,\infty)\to\mathbb{R}$ is a continuous function which satisfies $f(1)=1$ and: $$f(\frac1{x+1})\cdot x=f(x)-1$$ Does there exist such a function, if they do, are there infinitely many? And is ...
### Taylor series $\ln(2+x)$ centered at $x=2$
Taylor series $\ln(2+x)$ centered at $x=2$. Is the correct result $$y=\ln \left(4\right)+\sum _{n=1}^{∞}\frac{\left(-1\right)^n}{4^{\left(2^{\Large n}\right)}}\cdot \frac{\left(x-2\right)^n}{n!}\ ?$$ ...