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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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$(1-x)^2$ function

I had a question about checking whether $f(x)=(1-x)^2$ is completely monotonic. My argument is that it is not, because: It is not strictly monotonic on $[0, \infty)$. It does not satisfy the ...
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Monotonicity of $ f'$

Let $f(x)=\sin x-x+\frac {x^3}{6}$ and $g(x)=\cos x-1+\frac {x^2}{2}$ for $x\in \Bbb R$.Then How to prove that $f(x)\ge 0$ for all $x\ge 0$? From given function it is clear that $f'=g$ and $g'(x)=x-\...
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A monotone function must have a tangent ray that does not cross with the function

Consider a monotonically increasing and differentiable function $y=f(x)$ that passes through the origin. $\gamma=\{(x,y)|y=f(x)\}$ is the graph of $f$. Claim: there exists a ray $R$ such that $R\cap ...
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Check if 1-x is completely Monotonic function

I had a question about how can I check whether $f(x)=1-x$ is completely monotonic. Could somebody provide a simple example based on this function.
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Looking for a functional form $f(x_1, x_2, x_3)$, increasing in $x_1, x_3 $ and $x_1-x_2$

I need to construct a simple function with three elements: $x_1, x_2$, and $x_3$. $x_1$ and $x_2$ are variables between $0$ and $1$. $x_3$ is positive and larger than $1$ I need $f(x_1, x_2, x_3)$ be ...
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Proving a function is monotone

Let $n\in \mathbb{N}$, $u_1,u_2,\ldots ,u_n>0$ and I want to prove that the function $$p(\alpha)=\frac{\sum_{i=1}^n u_i^\alpha}{\left( \prod_{i=1}^n u_i^\alpha \right)^{1/n}}$$ is monotone in ...
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Proving concavity of derivative

Let $f(x)$ be defined and continuous and derivable for $x>-1$, $f(0)=1$, $f’(0)=0$ and $$f''(x) = \frac {1+x}{1+f(x)}.$$ Prove that $f’(x)$ is concave up for all $x>-1$. My attempt: I ...
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When is a positive matrix a monotone operator?

Let $A$ be an $n \times n$ matrix with all elements positive, and $\lambda$ its largest eigenvalue, which is real, positive, and of multiplicity one. Define $M = (1/\lambda) \, A$. Let $x$ be a ...
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If $f:[0,t]\to[0,t]$ is continuous and increasing, why can we conclude $f(0)=0$ and $f(t)=t$?

Let $t\ge0$ and $f:[0,t]\to[0,t]$ be continuous and (strictly) increasing. Why can we conclude $f(0)=0$ and $f(t)=t$? I've tried the obvious thing: Let $\varepsilon>0$. Since $f$ is increasing and ...
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Existence of striclty monotone transformation

Assume we have a function class $F$ containing bivariate functions $f(x,y)\; (f: \mathcal{X} \times \mathcal{Y} \to \mathbb{R})\ $ that are continuously differentiable with respect to each argument. ...
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Estimation of fractional expression

We define $$\displaystyle f(x,y)=\frac{1}{x^{2y}-\frac{1}{4^y}}+\frac{1}{(1-x)^{2y}-\frac{1}{4^y}} \text{ for } (x,y) \in \left[0,\frac{1}{2}\right) \times \left(\frac{1}{2},1\right]$$. A study ...
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Boundedness of derivative of bounded, monotonous, continuously differentiable function

Let $f\in C^1(\mathbb{R})$ be bounded and monotonous. What else do we need from $f$ for its derivative $f'$ to be bounded, too?
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When can we use derivative test to identify injective function?

I have doubt regarding first derivative test for identifying whether a function is injective or not: For example: $$f(x)=\ln x$$ has domain $(0, \infty)$. Now $$f'(x)=\frac{1}{x} \gt 0$$ hence $f(...
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One-sided limits of a monotonic function

Let $f: \mathbb{R} \to \mathbb{R}$ be an increasing function. I am trying to prove that for any 2 distinct points of discontinuity $a,b$ of this function, if $a<b$, then $f(a^+)<f(b^-)$. I have ...
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How to show that one not monotonous f doesn't have fixpoints?

I have a question about fixed points If I have one function $f$ (that is not monotonous!) I would like to demostrate that this function hasn't fixed points. I need to find a funciton $f$ for which ...
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If $\mathbb E[X]$ is increasing in $\theta$, is $\mathbb E[X^2]$?

Consider a discrete random variable $X\in[0,1]$ with p.m.f. $f_X(x)$ parametrized by $\theta$. Assume its expected value $\mathbb E[X]$ is increasing in $\theta$. Is $\mathbb E[X^2]$ increasing in $\...
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If $f(t)$ is continuous for $t$ $\in [0,1]$, $f' > 0$ and $f''(t) > 0$ for $t \in (0,1)$, do we have that $f'(t)$ is strictly increasing on $[0,1]$.

If $f(t)$ is continuous for $t$ $\in [0,1]$, $f' > 0$ and $f''(t) > 0$ for $t \in (0,1)$, do we have that $f'(t)$ is strictly increasing on $[0,1]$? Here is what i think: Since $f(t)$ is ...
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How to show this function is increasing both intuitively and using formula

In a book, it is claimed that the following function is obviously increasing as $x$ gets larger. I tried a few numerical examples for $x$, and it appears to be so. However, it is not clear to me how ...
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Does $f(x_n^+)=\inf\limits_{x_n<t< b}f(t)=\sum\limits_{x_i\leq x_n}c_i$ hold? Remark 4.31 in “Principles of Mathematical Analysis” by Walter Rudin.

Let $E$ be a countable subset of $(a, b)$. Let $\phi$ be a bijection from $\mathbb{N}$ to $E$. Let $\{x_n\}$ be a sequence such that $x_n := \phi(n)$. Let $\{c_n\}$ be a sequence of positive numbers ...
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Is ‎$‎f‎$‎ ‎monotone ‎when ‎$‎f‎$ ‎is ‎concave?‎

‎Let ‎$‎f:[1, +‎\infty‎)‎‎‎\rightarrow‎‎\mathbb{R}$ ‎be a‎ ‎concave ‎function. Suppose‎ $‎F:[1, +‎\infty‎)‎‎\rightarrow‎‎\mathbb{R}‎$ is a primitive function of ‎$‎f‎$‎. My ‎questions ‎are‎:‎ ‎‎ ‎(a) ...
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Construct a new function

I want to construct a function $f(x,y)$ ($x$ and $y$ are positive integers) which satisfies the following properties: 1) For each fixed $y$, (a) $f(x,y)<f(x+1,y)$ when $x<y$ (b) $f(x,y)>f(...
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Are there any equation that could produce a monotonic smooth step function by parameter

I want to write a mathematic formular that, given any number of monotonic arbitrary point, it will produce a monotonic smooth step function Such as a figure below, I give it 2 point (the intersect ...
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equation with exponential functions

Solve the following equation over the real numbers(preferably without using calculus): $$ 4^x + 4^{1/x} =18 $$ I already know the solutions thanks to Wolfram, what I have trouble with is proving ...
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How do I prove the following: Let f be increasing and bounded below on (a,b) with largest lower bound l. Then f(x) tends to l as x tends to a+.

Currently working my way through some proofs for monotone functions and am struggling with the proof stated in the question. I have proved: Let $f$ be increasing and bounded above on $(a,b)$ with ...
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Example of convex and injective $f:I \to \mathbb{R}$ such that $f^{-1}$ is not concave

Example of a convex and injective function $f:I \to \mathbb{R}$ on an interval $I$, such that $f(I)$ is an interval and $f^{-1}:f(I)\to \mathbb{R}$ is not concave. Attempt. Our function $f$ cannot be ...
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Prove if $\displaystyle{\lim_{n\to\infty}}f(n) = L$‎, then $\displaystyle{\lim_{x\to\infty}}f(x) = L‎$.

Let ‎$‎f:‎\mathbb{R}‎‎‎\rightarrow‎‎\mathbb{R}‎‎‎$ ‎be a‎ ‎monotone ‎function ‎such ‎that ‎‎$‎‎\displaystyle{\lim_{n\to\infty}}f(n) = L‎$‎. Then, $‎‎\displaystyle{\lim_{x\to\infty}}f(x) = L‎$‎.‎ ‎ I ...
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Dependence of sign of integral

suppose that $f,g : \mathbb{R}$ to $\mathbb{R}$ are differentiable functions such that f is strictly increasing and g is strictly decreasing. Define $p(x) = f(g(x)$ and $q(x) = g(f(x)),\forall x\...
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monotone function without derivative test

How i can prove this function is not a monotone function without the derivative test? $$f(x)=-\frac{1}{x^3}$$ thanks in advance
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Why does a monotonic function always have a positive rate of change?

Based on my book The rate of change of $f(u)$ as $u$ changes can be measured by looking at the change in $f$ between two values of $u$, divided by the change in $u$: $$\frac{Δf}{Δu} = \frac{f(u_2)...
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Lower bound $\mathcal{K}$-class function

Let $\alpha_0: \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}$ be a $\mathcal{K}$-function, i.e., a strictly increasing function such that $\alpha_0(0) = 0$, and $b \geq 0$ a given constant. Is there any ...
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Existence of a strictly increasing transformation between two functions [closed]

Assume $f$ and $g$ are two differentiable functions defined on a compact interval $X \subseteq \mathbb{R}$ mapping into $\mathbb{R}$ . I want to proof or disproof the following statement $ \forall x \...
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Every continuous open mapping from $\mathbb{R}$ into $\mathbb{R}$ is monotonic

Consider the image of an open set $(a,b)$ under the open and continuous mapping $f$. We show, $f$ cannot have any extremum in $(a,b)$. We know, connected sets are mapped to connected sets under a ...
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51 views

Monotonic equation

I have some equations like this: q1 = u1 * h1 , q2 = u2 * h2 , q3 = u3 * h3 (note ...
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1answer
87 views

Showing monotonicity for ratio of binomial pmf and tail cdf

I'm interested in showing for $X\sim\text{Bin}(n,p)$, $p\in(0,1)$ that when $x\geq np$, $$ \frac{P(X=x)}{P(X\geq x)}\leq \frac{P(X=x+1)}{P(X\geq x+1)} $$ I've verified using numerical simulations, but ...
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1answer
31 views

Properties of discontinuity of the second kind

Using Rudin's definition of a discontinuity of the second kind for a function. f has a discontinuity of the second kind if either $f(x^+)$ or $f(x^-)$ does not exist. Supposing that $f$ has a ...
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Prove monotonic function on $\mathbb{R}$ under given condition

Let $f(x)=e^{x}-x^2-ax$ (a) Prove when $a\leq 2-2ln(2)$ , $f(x)$ is monotonic function on $\mathbb{R},(a,+\infty)$ (b) Given when $x>0$, $f(x)\geq 1-x$ always true. Find the range of $a$. This ...
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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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1answer
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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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1answer
40 views

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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1answer
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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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55 views

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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2answers
85 views

$x_n=1+\frac{1}{\sqrt2}+…+\frac{1}{\sqrt n}-2\sqrt n$

$x_n=1+\frac{1}{\sqrt2}+...+\frac{1}{\sqrt n}-2\sqrt n$ 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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2answers
59 views

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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1answer
51 views

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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46 views

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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4answers
124 views

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) $...