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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.

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Convergence of monotone nets

In sequences of real numbers, we have a monotone convergence result: If $a_{n+1}\geq a_n$ and bounded, then $a_n$ converges to it's supremum. The proof seems to work also in the net case. My ...
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Intermediate Value Theorem Proofs

Today was the first day that I was introduced to the intermediate value theorem and I'm still quite unsure on how to use it to help solve some proofs. I've been given the following proofs to take ...
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the product of two completely monotone functions

A function $f:(0,∞)→[0,∞]$ is said to be completely monotonic if its $n$-th derivative exists and $(−1)^nf^{(n)}(x)≥0$, where $f^{(n)}(x)$ is the $n$-th derivative of $f$. Given two completely ...
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Let $f$ be a continuous monotone function. Show that $f$ must be absolutely continuous on [0,1]

Let $f:[0,1]\to\mathbb{R}$ be a continuous monotone function such that $f$ is differentiable everywhere on (0,1) and $f'(x)$ is continuous on $(0,1)$. Show that $f$ must be absolutely continuous on $[...
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How to prove $a_{n+1}=(1+a_n+a^2_{n-1})/3$ is a non-decreasing sequence?

$$a_1=a_2=0.5$$ It isn't hard to show that $0.5\le a_n\lt 1$, and that if the sequence converge, the limit is 1. But how to prove it's monotone? I've tried: $$a_{n+1}-a_n=\frac{1-2a_n+a^2_{n-1}}{3}\...
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If two functions are increasing is their product increasing?

The problem is: If $f$ and $g$ are increasing, then is $f \cdot g$ also increasing? First, I started out by working some basic definitions and assumptions: Assume $a < b$ Increasing means $f(...
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Why are the local extrema of a log-transformed function equal to local extrema of the original function?

I am studying maximum likelihood and to simplify taking the derivative of the likelihood function, it is often transformed by the natural log before taking the derivative. I have read in other posts ...
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Existence of Totally Non-monotonic Continuous Real-valued Function in $[0, 1]$

Definition: Let $f: [0, 1] \rightarrow \mathbb{R}$ be a function. Say $f$ is Totally-Nonmonotonic iff for any $a, b$ in [$0, 1$], $f$ is NOT monotonic on [$a, b$]. Question: Let $f: [0, 1] \...
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Monotonicity of $a_n=1+\frac{(-1)^n}{n}$

I'm trying to study the monotonicity of $a_n=1+\frac{(-1)^n}{n}$, but what I'm getting isn't correct: I just assume that $a_n$ is monotonically increasing, and if it isn't, I'll get something absurd: ...
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$x_n=1+\frac{1}{\sqrt2}+…+\frac{1}{\sqrt n}-2\sqrt n$

$x_n=1+\frac{1}{\sqrt2}+...+\frac{1}{\sqrt n}-2\sqrt n$ From here Investigating the convergence of a series using the comparison limit test, Part II I can see that the sequence converges. I was ...
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Limits of monotone function

What does the notation $f(x^{+})$ and $f(x_+)$ mean? The context is the following I have a proposition concerning monotonic increasing functions, so $f$ is nondecreasing, also $x\in(a,b) =I$ where $f$...
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Proving the complete monotonicity of $f$ given $(-\log f)'$ is completely monotone

A function $f:(0,∞)→[0,∞]$ is said to be completely monotonic if its $n$-th derivative exists and $(−1)^nf^{(n)}(x)≥0$, where $f^{(n)}(x)$ is the $n$-th derivative of $f$. Prove that if $(-\log f(...
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How to find the $\lim_{n\to\infty}s_n$ when $s_1=5, s_n =\sqrt{2+s_{n-1}}$?

How to find the $\lim_{n\to\infty}s_n$ when $s_1=5, s_n =\sqrt{2+s_{n-1}}$ using the Monotone Convergence Theorem? I have the proof from my professor, but I am stuck at one step. Proof: Step 1: ...
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If $f$ be a decreasing function satisfying $f(x+y)= f(x)+ f(y)- f(x)f(y) ~\forall x, y \in \mathbb R$ and $f'(0)= -1$

If $f$ be a decreasing function satisfying $f(x+y)= f(x)+ f(y)- f(x)f(y) ~\forall x, y \in \mathbb R$ and $f'(0)= -1$ then $\displaystyle\int_0^1 f(x)dx $ is: A)$1$ B) $1- e$ C) $...
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Betweenness preserving implies monotonic?

For this question, we can assume that $f:\mathbb{R}\rightarrow\mathbb{R}$. However, I hope that an answer can generalize to arbitrary linearly ordered sets. I assume that everyone will know what I ...
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Prove the sequence $a_{1} = 4$, $a_{n + 1} = \frac{a_{n}}{2} + \frac{2}{a_{n}}$, $n = 1, 2, \ldots$ satisfies $a_{n} > 2$

Prove the sequence $a_{1} = 4$, $a_{n + 1} = \frac{a_{n}}{2} + \frac{2}{a_{n}}$, $n = 1, 2, \ldots$ satisfies $a_{n} > 2$ Let $x = a_{n}/2$. Then $a_{n + 1} = x + 1/x$. Define $f(x) = x + 1/x$ ...
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Monotonicity of tangent

obviously for $a,b \in ]\frac{-\pi}{2},\frac{\pi}{2}[$ the ordinary tangent map is strictly monotonic. Hence for $b \geq a \Rightarrow \tan(b) \geq \tan(a)$. In the proof I try to understand it ...
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How to find CDF of $Y=|X|\wedge 2$ with $X\sim Laplace(\lambda)$

Given $X$ a bilateral exponential with density $f_X(x)=\frac{1}{2}e^{-|x|}, \forall x\in \mathbb{R}$ and $\lambda=1$, i have to find CDF of $Y=|X|\wedge 2$. I know that $Y$ is not a monotonic ...
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Prove inverse of strictly monotone increasing function is continuous over the range of original function

Let $f:[a,b] \rightarrow \Bbb R$ be a strictly monotone increasing. Then $f$ has an inverse function $g:[c,d]\rightarrow \Bbb R,$ where $[c,d]$ is the range of $f$. I'm trying to prove that $g$ is ...
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Monotonicity of function $f(x)=(1+\frac1x)^x(1+x)^\frac1x$

Given function for $x>0$ $$f(x)=(1+\frac1x)^x(1+x)^\frac1x$$ which is not a monotonic function, but it is easy to find the only maxima $$f(1)=4$$ so, can we find a strict prove showing $f(x)$ ...
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Prove $a_n = \sqrt{n+1}-\sqrt{n}$ is monotone $n \ge 0$.

Prove $a_n = \sqrt{n+1}-\sqrt{n}$ is monotone $n \ge 0$. To be monotone it must be either increasing or decreasing, so: $a_n \ge a_{n-1}$ or $a_n \le a_{n-1}$ $\sqrt{n+1}-\sqrt{n} \ge? \sqrt{n}-\...
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Is the function $f(x)=\frac{\sin(\sin{(x)}+x)}{2+\cos{(\lvert x\rvert}+\cos{(x)})}$ monotonic and/or periodic?

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ $$f(x)=\frac{\sin(\sin{(x)}+x)}{2+\cos{(\lvert x\rvert}+\cos{(x)})}$$ I am having trouble showing whether or not this function is monotonic and/or periodic. ...
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Are all linear functions monotonic?

I have never come across a statement linking linearity and monotony - but it seems that for each linear function (positive, negative or even constant slope), the function is monotonic: I.e. for y $\...
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Prove or disprove: If $f$ is increasing and differentiable on $(a,b)$ then $f'(x)\ge 0$ on $(a,b)$

Here's the question again: Prove or disprove: If $f$ is increasing and differentiable on $(a,b)$ then $f'(x)\ge 0$ on $(a,b)$. I could not find any counterexamples to it so here's my attempt at ...
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Prove $\{|C_a^n|\}$ is decreasing starting from some $k$ given $C_a^n = \frac{a(a-1)(a-2)\dots(a-n+1)}{n!}$

Let $C_a^n$ be defined as: $$ \begin{cases} C_a^n = \frac{a(a-1)(a-2)\dots(a-n+1)}{n!}\\ n\in \mathbb N \\ a \in \mathbb R \\ C_a^0 = 1 \end{cases} $$ Prove that $\{|C_a^n|\}$ is decreasing ...
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Any countable set of real numbers is set of discontinuities of some monotone function.

I am studying for a final exam and have come across the following old exam question: Prove that any countable set of real numbers is the set of points of discontinuity of some monotone function. The ...
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An Example of a Non-continuous Injection on a Real Interval, that is Not Strictly Monotonic.

Theorem Let $I$ be a real interval. Let $f:I \to \mathbb R$ be an injective continuous real function. Then $f$ is strictly monotone. If the condition on continuity indeed is necessary (...
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Prove that $C_{3 \over 2}^n$ is bounded given $C_{a}^n = \frac{a(a-1)(a-2)\dots(a-n+1)}{n!}$

Let: $$ \begin{cases} C_{a}^n = \frac{a(a-1)(a-2)\dots(a-n+1)}{n!}\\ C_{a}^0 = 1 \end{cases} $$ Prove $C_{3 \over 2}^n$ is bounded. I've started with finding a reduced formula: $$ C_{3\over 2}^n ...
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What does it mean for a function to be semi-monotonic?

I mostly understand monotonic functions as described by wikipedia. However, I do not understand what it means for a function to be semi-monotonic as described in the java math class. This page helped ...
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Proof of Froda's Theorem (explanation)

Theorem: Let $f$ be a real valued function of real variable defined on open interval $(a,b)$ and let $f$ be monotonic. Then the set of all discontinuities is at most countable. I would like an ...
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Verify proofs related to monotonicity of $x_{n+1} = {1\over 2}(x_n+y_n)$ and $y_{n+1} = \sqrt{{1\over 2}(x_n^2 + y_n^2)}$

Let $\{x_n\}$ and $\{y_n\}$ be sequences defined by recurrence relations: $$ \begin{cases} x_{n+1} = {1\over 2}(x_n+y_n)\\ y_{n+1} = \sqrt{{1\over 2}(x_n^2 + y_n^2)} \\ x_1 = a > 0\\ y_1 = b > ...
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Show that $x_{n+1} = \frac{2+x_n^2}{2x_n}$ is a decreasing sequence.

Let $x_n$ be defined as: $$ \begin{cases} x_{n+1} = \frac{2+x_n^2}{2x_n} \\ n\in \mathbb N \\ x_1 = 4 \end{cases} $$ Show that $x_n$ is a decreasing sequence. I'm having a hard time with the ...
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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+...
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Show $x_{n+1} = {1\over 2}x_n^2 - 1$ is bounded below and unbounded above and $x_n$ is increasing.

Let: $$ \begin{cases} x_{n+1} = {1\over 2}x_n^2 - 1\\ x_1 = 3\\ n\in \mathbb N \end{cases} $$ Show that the sequence $x_n$ is bounded only below and is increasing. I've started with the ...
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Show that $x_{n+1} = \frac{x_n + 1}{n+1}$ is decreasing starting from some $n_0$ and find $n_0$.

Given $n\in \mathbb N$ and: $$ \begin{cases} x_{n+1} = \frac{x_n + 1}{n+1} \\ x_1 = -10 \end{cases} $$ Show that $x_n$ is decreasing starting from some index $n_0$. Find $n_0$. I've tried to ...
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Why $g(t)=f(x+t(y-x))$ is non-decreasing when $\langle \nabla f(y)- \nabla f(x),y-x \rangle \geq$ 0?

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be a differentiable function. Now define the differentiable univariate function as follows $$g(t)=f(x+t(y-x)), \,\,\, 0 \leq t \leq 1 $$ where $x,y \in \...
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Show that for $a \ne 1$, $a > 0$ the sequence $\{x_n\} = n(1-a^{1\over n})$ is increasing

I'm having difficulties with the following problem: Let: $$ \begin{cases} x_n = n(1-a^{1\over n})\\ a > 0 \\ a \ne 1 \\ n \in \mathbb N \end{cases} $$ Show that $\{x_n\}$ is an increasing ...
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Show that $x_n = \left(1+{x\over n}\right)^{n+k}$ is decreasing for $0<x<k$; $n, k\in \mathbb N$

I've been working on some classical proofs of the sequences in the form $\left(1+{x\over n}\right)^p$. So: Let $n,k \in \mathbb N$ and: $$ \begin{cases} x_n = \left(1+{x\over n}\right)^{n+k}\\ 0 ...
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Monotone Increasing Concave Function

Given monotone increasing concave function $f(x):\mathcal{R}_{\geq 0} \to \mathcal{R}_{\geq 0}$, Can we say that $$ f(d_1)+f(d_2)-f(d_1+d_2) \leq f(d_3)+f(d_4)-f(d_3+d_4) $$ if $d_1<d_3$ and $d_2&...
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Proof verification of $\{x_n\} = \left(1 + {1\over 2n}\right)^n$ is an increasing sequence.

Let $n\in \mathbb N$ and: $$ x_n = \left(1 + {1\over 2n}\right)^n $$ Show that $\{x_n\}$ is an increasing sequence. $\Box$ Consider ratio test of two consequent terms $x_n$ and $x_{n+1}$: $$ \...
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Proof verification of if $\{x_n\}$ is monotone then $\{y_n\} = {1\over x_1 + x_2 + \dots + x_n}$ is monotone

Let $n\in \mathbb N$ and $\{x_n\}$ is a monotone sequence. Prove that: $$ \{y_n\} = {1\over x_1 + x_2 + \dots + x_n} $$ is also a monotone sequence. Given $\{x_n\}$ is monotone then by ...
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Verifying an increasing function on a closed interval is Riemann Integrable

So I was having some difficulty coming up with a conceptual reason for why an increasing function on a closed interval would be Riemann Integrable, when without effort a proof seemingly fell out of ...
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partial derivative with respect to ratio of two variables

This may be a pretty nonsensical question. But I will appreciate if I can an answer (I think I know what I am doing is right, but still want to confirm. Given the following expression, can I take ...
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1answer
41 views

Find relation between $a,b,c,d$ such that $\{x_n\} = \frac{an + b}{cn+d}$ is increasing/decreasing starting from some $n_0$.

I'm trying to solve the following: Let $n\in \mathbb N$ and $\{x_n\}$ be a sequence defined by: $$ \{x_n\} = \frac{an + b}{cn+d} $$ Find the relation between $a,b,c,d$ such that $\{x_n\}$ is ...
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1answer
38 views

Prove $x_n = \frac{n-3}{\sqrt{n^2+1}}$ is monotonic starting from some $n_0$.

I'm having some difficulties with the following problem: Let $n\in \mathbb N$ and: $$ x_n = \frac{n-3}{\sqrt{n^2+1}} $$ Prove $x_n$ is a monotonic sequence starting from some $n_0$. I've ...
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6 views

How to prove monotonicity of symmetric positive definite quadratic form

Let $a=(a_1,...,a_n) \in \mathbb{R}^n$, $a_1>0$ and $h>0$. Define $b=(a_1+h,...,a_n)$. Furthermore let $C$ be a positive definite symmetrix matrix How can i proof that $b^TCb>a^TCa$?
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On the monotonicity of a certain sequence

Consider the sequence $a_n = \left( 1+ \frac{1}{n} \right)^{n + k}$, where $ 1\leq n \in \mathbb{N}$. Then the sequence is decreasing for $k \geq \frac{1}{2}$, and increasing for $k < \frac{1}{2}$. ...
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If $f$ is monotonic on $(a,b)$, the set of points of (a,b) at which $f$ is discontinuous is at most countable.

Now as an undergraduate student, I am studyign baby Rudin. I know the proof of this theorem are already well explained on match stack exchange here, but I have some question about the proof. In page ...
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If f is a monotone function which defined on interval, then f is measurable [duplicate]

Prove that If f is a monotone function which defined on interval, then f is measurable. If f is increasing and define on interval, then the set A={x:f(x)>a} will be an interval for all a, and it's ...
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Prove sequence $x_n = \frac{3n+5}{\sqrt{4n^2 - 1}}$ is bounded.

Let $n \in \mathbb N$ and: $$x_n = \frac{3n+5}{\sqrt{4n^2 - 1}}$$ Prove $x_n$ is a bounded sequence. How can I show that the sequence is bounded? I was thinking about the monotonicity and ways to ...