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Questions tagged [monotone-functions]

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. This tag may also include questions about applications or consequences of monotonicity, such as convergence, optimization, or inequalities.

In calculus, a function $f$ defined on a subset of the real numbers with real values is called monotonic if and only if it is either entirely increasing or decreasing. It is called monotonically increasing (also increasing or nondecreasing), if for all $x$ and $y$ such that $x \leq y$ one has $f(x) \leq f(y)$, so $f$ preserves the order. Likewise, a function is called monotonically decreasing (also decreasing or nonincreasing) if, whenever $x \leq y$, then $f(x)\geq f(y)$, so it reverses the order.

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The function $f (n) = (1 + 1 / n) ^ {n+1}$ is decreasing

I cannot prove that the function $$f (n) = \left(1 + \frac1n\right) ^ {n + 1},$$ defined for every positive integer $n$, is strictly decreasing in $n$. I already tried to prove by induction and also tried to prove by calculating the difference…
Paulo Argolo
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Monotonicity of ratio of sums $\frac{\sum f_i}{\sum g_i}$ when each $\frac{f_i(x)}{g_i(x)}$ is increasing

Suppose that I have a sequence of functions $f_i$ and $g_i$, and I know that for each $i, \frac{f_i}{g_i}$ is increasing in its argument. Is it then also true that $\frac{\sum_i^N f_i(x)}{\sum_i^N g_i(x)}$ is increasing in $x$? Additionally I know…
Bayesian
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Monotonicity finding based on given releationships

I have a function $f:\mathbb{R}\rightarrow\mathbb{R}$ with $f''(x)>0\text{ }\forall x\in\mathbb{R}$ and $\lim_{x\to+\infty}{f(x)}=0$. Now I want to prove that $f$ is strictly decreasing. I tried proving this with reductio ad absurdum but I stack…
Leos Kotrop
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Finding nondecreasing function where $xf(x)+(1-x)f(1-x)=1$

I've been wracking my brain for a couple of days without making much progress trying to find a function $f(x):\mathbb{R} \rightarrow \mathbb{R}$ with the following properties. $f(0)=0$ $f(1)=1$ When $0\leq x\leq 1$: $f(x)$ is continuous…
Bob
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How to show this implicit function is decreasing?

Let $f(x, y)\equiv(y-x)\left(\frac{y-1}{y}\right)^{1-x}$, where $0\hat{y}(x)$. Here $\hat{y}(x)\equiv\arg max_y [f(x, y)-x]$ is the unique peak of $f(x, y)$ as a function of $y$. The implicit function $y=y(x)$ is given by $$f(x, y)=k\cdot…
zmxm
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common fixed point of commuting functions of unit interval

Let $f,g:[0,1]\to[0,1]$ commute ($f\circ g=g\circ f$). Suppose $f$ is weakly increasing and $g$ is continuous. Is it necessarily true that there exists $x\in[0,1]$ such that $x=f(x)=g(x)$ ? Motivation. $g$ necessarily has a fixed point by mean value…
tomm
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Prove that the function $ g $ satisfying $g(g(x))=2g(x)-x $ is strictly monotonic.

Let $ g :\Bbb R \to \Bbb R$ a continuous function such that $$(\forall x\in \Bbb R)\; g(g(x))=2g(x)-x$$ Prove that $ g $ is injective and strictly monotonic. I took $ x,y\in \Bbb R $. $$g(x)=g(y)\implies g(g(x))=g(g(y))$$ $$\implies 2g(x)-x=2g(y)-y…
hamam_Abdallah
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Vector spaces of monotone functions

This might be quite trivial and I may be missing something obvious. Let $V$ be a vector space of functions from $\mathbb R$ into itself such that every function in $V$ is monotonic. Does it follow that the dimension of $V$ is at most $2$? $\{ax+b:…
Kavi Rama Murthy
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The image of a 'pathological' function.

This question is motivated by this another one. Let be some convergent series of positive terms $\sum_{k=1}^\infty a_k$ and some enumeration $\{x_k\}_{k=1}^\infty$ of $\Bbb Q$. Define for $x\in \Bbb R$, $$f(x)=\sum_{x_k
ajotatxe
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Proof that function is increasing

How to prove that the function $$y(x)=x(\ln(x+1) - \ln(x))$$ is increasing on $[0,1]$? The derivative test requires to analyze equally challenging function $\ln{\left(\frac{x+1}{x}\right)}-\frac{1}{x+1}.$ Are there more ways to prove that $y(x)$ is…
user314849
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continuity of monotonically increasing function

Let $f:[0,1]\to[0,1]$ be monotonically increasing. Which of the following statements is/are true? $f$ must be continuous at all but finitely many points in $[0,1]$ $f$ must be continuous at all but countably many points in $[0,1]$ $f$ must be…
dipali mali
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Prove that $f(n) =\frac{(n+1)^n}{n^{n+1}}$ is Monotonic

Prove that for each $n \in N$, $$f(n) =\frac{(n+1)^n}{n^{n+1}}$$ is monotonic. First, I can tell that the function is decreasing. If I take $\frac{1}{n}$, the function looks like $\frac{1}{n}(1+\frac{1}{n})^n$. Can this help?
iTayb
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How to prove a function is strictly increasing?

Let $f(x)$ be a continuous strictly increasing smooth function on $[0,1]$ with $f(0)=0$ and $f(1)=1$. In addition, $F(t)=\int_0^t f(x)dx$ and $\phi>1$. Prove that the following function $h(t)$ is strictly increasing with respect to $t$ on $(0,1)$,…
cclinoom
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How to prove monotonicity of $\exp(x)-\sin(\exp(-x))$?

I want to prove that $x< y \implies f(x) < f(y) \forall x,y \in [0,\infty)$ for the function $f(x)=\exp(x)-\sin(\exp(-x))$ without using derivatives. I know that $e^x$ is increasing but $\sin(e^{-x})$ is decreasing and $\sin(e^{-y})< \sin(e^{-x})…
Quaeram
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If $f(x)$ and $f(x)g(x)$ are strictly increasing and positive, is $g(x)$ also strictly increasing?

Suppose we have two functions $f(x) \in (0,1)$ and $g(x) \in (0,1)$ such that $f(x)$ and $h(x) = f(x)g(x) \in (0,1)$ are both strictly increasing on the support of $x$. Can I claim that $g(x)$ is an increasing function?
pineapple
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