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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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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 come it is so obvious. Could anyone point out the…
LaTeXFan
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How to prove inequality with stricly increasing concave function?

I have following inequality I need to prove: $$\frac{x_i +x_j}{x_i}>\frac{u(R)}{u(R\frac{x_i}{x_i +x_j})}$$We have that $x_i,x_j,R>0$ as well as $u$ being an strictly increasing concave function in the respective interval.
MyloRen
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Monotonic function is rectificiable

Let $f: [0,1]\to\mathbb{R}$ be a monotonic function. Show that $f$ is rectificiable. What is the length of $f$? At first I had to show, that $f$ is at at most countable points not continuous. Now I struggle to show, that $f$ is rectificiable. I…
Cornman
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Monotonicity of sum of exponential functions

Let $f(x)=2^x+5^x-7^x$. The problem is to study the monotonicity intervals of $f(x)$. My first attempt to tackle the above, is to study the sign of the derivative of $f(x)$ $f^\prime (x)=ln(2)\cdot 2^x+ln(5)\cdot5^x-ln(7)\cdot7^x$ However I can see…
D. Vaf
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Check of Monotone Property

I would like to check whether $f: \Bbb [-\pi/2, \pi/2] \in \Bbb R \rightarrow \Bbb R$ where $f(x) = \tan x$ is monotone increasing function. Of course it looks fairly true in my bare eye with denoted in the printed version of textbook, however, want…
Daschin
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Induction proof: monotonic inreasing

Let c:$\mathbb{N}\rightarrow\mathbb{N}$ be a strictly monotonic inreasing function, proof by induction that $c \geq id_{\mathbb{N}}$ where id is the identity $\mathbb{N} \rightarrow \mathbb{N}:n \rightarrow n$. I have to do it with induction, but c…
user337258
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What are the real life examples of monotonic functions?

Please tell me what are the real life examples of monotonic functions? One example for increasing functions and one for the decreasing functions.
Umair Nadeem
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Strictly monotonic increasing function lower bound

For all strictly monotonic increasing function $f:[0, \infty) \rightarrow [0,\infty)$ such that $f(0)=0$ and $lim_{x\rightarrow \infty} f(x)=\infty$, I would like to know if there always exists another strictly monotonic increasing function $f_2:[0,…
mgdavo
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What does this symbol mean in this context (#)

Let D ⊆ R and let f:D→R. Assume that #D ≥ 4. Assume that f is strictly 4-monotone, i.e., assume, for all S ⊆ D, that [ #S = 4 ] ⇒ [ (f|S) is strictly monotone ]. Show that f is strictly monotone.
samsonite
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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
Feemo Fellow
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Is k=$\sqrt{6+\sqrt{6+...}}$ monotonically increasing?

Is k=$\sqrt{6+\sqrt{6+...}}$ monotonically increasing or is not ? i know it converges to $3$.
Gibberish
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Prove that $\sin^4n/\sqrt n$ is monotonically decreasing

How can we prove that this function $$f(x) = \frac{\sin^4x}{\sqrt x}$$ is monotonically decreasing? I tried to use usual method using derivative, but it do not give us an answer.
niar_q
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