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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.

1235 questions
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Prove that $\frac{x} {x+1}$ is increasing

I am trying to prove this function $y = \frac{x} {x+1}$ is increasing. I tried doing by induction: let $n = 1$, then clearly $\frac 1 2 \leq \frac 2 3$ so it's true. But I get stuck on the inductive step, where I'm not even sure how to manipulate…
user843046
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monotonicity of $\frac{f(x)}{x}$

If $f$ is defined on $[0,\infty)$, and $x_1,x_2>0$, if $\frac{f(x)}{x}$ is monotonic increasing, prove that $f(x_1+x_2)\geq f(x_1)+f(x_2)$. I suppose it has something to do with starting from $x_1\leq x_1+x_2$ and $x_2\leq x_1+x_2$, but I can't…
Lee Laindingold
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Is this claim true about the monotonicity of $\frac{\cosh 2 x^3 }{3\cosh 5 x^3 }$?

I want to check the monotonicity of the function for $x>0$ $$\frac{\cosh 2 x^3 }{3\cosh 5 x^3 }$$ Computing the first derivative, it can be proved that it is negative and then the function is decreasing. My question is can we claim that since…
charmin
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Describe monotonicity of $f(x)=|\log (x+1)|$ at $x=0$

The function is $$f(x)=\begin {cases} \log (x+1) & x\ge 0 \\ -\log(x+1) & x <0 \end {cases}$$ So $$f’(x)=\begin{cases} \frac{1}{x+1} & x\ge 0 \\ \frac{-1}{x+1} & x<0 \end {cases}$$ Now $f’(0) = 1 > 0$, so it should be strictly increasing From the…
Aditya
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Two related questions on monotonicity and use of the derivative

Let $w:\mathbb{R}^m\rightarrow\mathbb{R}^m$ a strongly monotone map, that is, there exists a $\gamma>0$ such that $$ [w(x+h)-w(x)]^\top h \geq \gamma\cdot \vert\vert h\vert\vert ^2. $$ I have a couple of questions then. First, let…
Daniele
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Prove that $f(x)=\sum_{i=0}^{M-1}(-1)^i\binom{M}{i+1}\frac{1}{1+ix\Delta}$ increases with $x$?

How to prove that $f(x)=\sum_{i=0}^{M-1}(-1)^i\binom{M}{i+1}\frac{1}{1+ix\Delta}$, where $\Delta>0,M>1$, increases with $x$?
Dave
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Verify the monotony of f without the use of derivative

$f:\Bbb R \rightarrow (2,+\infty)$ and it is known that $$f^3(x)-12f(x)=3x-15$$ I need to verify f's monotony but without the use of the derivative so Let there be a second function $$H(x)=x^3-12x \qquad \forall x \epsilon (2,+\infty)$$ so we got…
Antonis Sk
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Monotonic relationship with plateaus and vertical portions?

For a monotonic relationship between $x$ and $y$, an increase $x$ always leads to an increase in $y$ or always leads to a decrease in $y$. On a graph, the slope is always positive or always negative. What is an almost monotonic relationship called…
user2153235
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Monotonic family of monotonic functions

I am imagining a family of monotonic functions, which can be represented as a parametric function $f(\cdot; \tau): \mathbb R \to \mathbb R$ parameterized by $\tau \in \mathbb R$ such that (1) when $\tau > 0$, $f$ is increasing, (2) when $\tau < 0$,…
Vezen BU
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Does this guarantee invertibility in higher-dimensional functions?

Let $f(\mathbf{x}):\mathbb{R}^N\mapsto\mathbb{R}^N$ be structured as: $$ f(\mathbf{x})= \left[\begin{matrix} f_1(x_1,...,x_N) \\ f_2(x_1,...,x_N) \\ \vdots \\ f_N(x_1,...,x_N) \\ \end{matrix}\right] $$ where each…
J.Galt
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Can we use equivalence symbol

we want to show that : $ f(x) = e^{-x} -x -1 $ is strictly decreasing. For all $x\in \mathbb{R}$ with : $ \left.\begin{matrix} x e^{-y} \\ x-y-1 \\ \end{matrix}\right\} \Leftrightarrow f(x)>f(y) $…
user1051034
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Difference of monotonic functions

Consider the strongly monotonic function $w:\mathbb{R}\rightarrow [a\;b]$, with $a,b\in\mathbb R$, that is, there exist a scalar $\eta(x,h)$ that verifies $$ \frac{w(x+h)-w(x)}{h}\geq \eta(x,h)>0, \quad \forall x,h\in\mathbb{R}. $$ I was wondering…
Daniele
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Find the solution(s) of the equation

Find the solution(s) of the following equation: ${\ln(2-x)}+1=2-e^{x-1}$, $x=1$ is a solution of the equation but I cannot show that it is the only solution. I find it diffucult to show that function $f(x)={\ln(2-x)}+e^{x-1}-{1}$ is monotonic.
Domesticus
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What are examples of discrete functions that are monotonically increasing/decreasing?

What are examples of discrete functions that are monotonically increasing/decreasing? Can monotonically increasing/decreasing be defined over discrete functions?
mallea
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Is a one-one non-decreasing function always increasing?

Suppose $f:[a,b]\rightarrow[c,d]$ is one-one and non-decreasing, then $f$ is always increasing. Is this statement correct? If not, give an example. The sketch of the proof is as follows: If any flaw in this, please point out. Now, Let $x,y \in…
panch
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