Tagged Questions

Questions on proving, manipulating and applying inequalities.

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Clarification on inductive proof of Bernoulli's inequality

Prove that if $h > -1$, then $1 + nh ≤ (1+h^n)$ for all nonnegative integers $n$. I've read several solutions and I'm still totally lost on how to go about this. I have the inductive hypothesis:...
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How to prove the inequalities between $20^{70^2},30^{60^2},40^{50^2}$

Let $$M=\{ 20^{70^2}, 30^{60^2},40^{50^2}\}$$. What number is the greatest and which is the smallest? I thought about beginning by assuming certain inequalities and trying to prove them, for example: ...
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Lets consider the following inequality form this question: $$|y^2-y-2|\geq 4 + |y^2+y-2|+|y+4|+|y|$$ User Did said that: $$|y^2-y-2|=|4+(y^2+y-2)-(y+4)-y|.$$ Using the triangular inequality ...
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A clean proof of $x^2 \geq x$, for any integer $x$

I am trying to prove that $x^2 \geq x$ for any integer $x$. Since we know that for any number $n$, $n^2 \geq 0$ we conclude that if $x \leq 0$ the proposition will hold. Next we must prove that the ...
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Is there any clever way to solve inequalities like $|y^2-y-2|\geq 4 + |y^2+y-2|+|y+4|+|y|$?

I have to solve different types of inequalites of this type: $$|y^2-y-2|\geq 4 + |y^2+y-2|+|y+4|+|y|$$ I know the standard method for solving these inequalities, by finding the all zeros of the ...
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How to solve this inequality problem?

Given that $a^2 + b^2 = 1$, $c^2 + d^2 = 1$, $p^2 + q^2 = 1$, where $a$, $b$, $c$, $d$, $p$, $q$ are all real numbers, prove that $ab + cd + pq\le \frac{3}{2}$.
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Why are generalized inequalities defined over proper cone?

Why generalized inequality is defined over a proper cone? What property does not hold if we define it over non-convex cone? Same with `pointed'. For example, generalized inequality makes sense in a ...
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Why do we set both factors x>0 when solving an inequality?

Here's a page from my math textbook: (http://s31.postimg.org/u2ow52xgb/Whats_App_Image_20160715.jpg) (unfortunately I can't post pictures due to low reputation) Why do we consider both factors as ...
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Proof of the square root inequality [duplicate]

I stumbled on the following inequality: For all $n\geq 1,$ $$2\sqrt{n+1}-2\sqrt{n}<\frac{1}{\sqrt{n}}<2\sqrt{n}-2\sqrt{n-1}.$$ However I cannot find the proof of this anywhere. Any ideas how to ...
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Probability in a inequality

I've been practicing probability for a while and now I encountered a math problem which I don't know how to approach. One integer is selected randomly from the set $[1,50]$ . What is the ...
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Find $x$ where $|(x-3)^2-1|\geq |(x+2)^3+5|$ [closed]

Find all real $x$ such that the following is true: $|(x-3)^2-1|\geq |(x+2)^3+5|$
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Is this inequality provable? $e^{\left(\pi^{e^{\pi^{.^{.^{.^{e^\pi}}}}}}\right)}\ge \pi^{\left(e^{\pi^{e^{.^{.^{.^{\pi^e}}}}}}\right)}$

I am interested in proving the following inequalities: $e^\pi\ge\pi^e$, $\quad \pi^{(e^\pi)}\ge e^{(\pi^e)}$, and $\quad e^{(\pi^{(e^\pi)})}\ge \pi^{(e^{(\pi^e)})}.$ How we can prove these ...
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system of inequalities in Mathematica

I am trying to solve the following system of inequalities in Mathematica but the output is just the same system of inequalities. I need to get the expressions for x, y in terms of the pi indexed ...
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Bounds on a system of coupled ODEs

Suppose we have a $1$-dimensional differential inequality $$\frac{dx}{dt} \leq x - x^3$$ We can apply the Comparison principle to claim that if $y(t)$ is the solution to $\frac{dy}{dt} = y - y^3$, ...
compare $x^{y^{z}}$ and $y^{x^{z}}$ (help)
the problem goes like this : let $(x,y,z) \in \mathbb{R^3}_{+}$ and $0<x<y<z$ compare $x^{y^z}$ ;$x^{z^y}$; $y^{x^z}$ ; $y^{z^x}$ ; $z^{y^x}$ ; $z^{x^y}$ how can we compare the values ?
Prove that for any $x_1,\dots,x_n>0$  {\root{n}\of{\prod _{k=1}^{n}\ \sum_{t=1}^{k}\ \frac{1}{t^2\cdot\sqrt[t]{x_1\cdot\ldots\cdot x_t}} }} \ \cdot\ \sum _{k=1}^{n}\frac{\sum_{j=1}^{k}\sum_{i=1}^...