# Tagged Questions

Questions on proving, manipulating and applying inequalities.

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### Prove that $|x-a_1|+|x-a_2|+|x-a_3|\geq a_3 - a_1$, for $a_1<a_2<a_3$, and determine the condition for equality.

I got this question from the first chapter of Courant and John's Introduction to Calculus and Analysis I. The problem is as follows: Prove that $|x-a_1|+|x-a_2|+|x-a_3|\geq a_3 - a_1$, for ...
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### Bounding the variance of a sum of independent random variables

Suppose $\{X_i\}_{i=1}^n$ is a sequence of independently distributed random variables that take values in $[0,1]$. Let $\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i$ denote the average of the sequence. ...
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### Why is Mathematical Induction used to prove solvable inequalities?

As a first year undergrad student I've seen problems where solvable inequalities need to be proven to hold in a specific domain using Mathematical Induction. My question is, if the inequalities are ...
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### If $abcd=1$,where $a,b,c,d$ are positive reals,then find the minimum value of $a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd$.

If $abcd=1$,where $a,b,c,d$ are positive reals,then find the minimum value of $a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd$. Let $E=a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd=(a+b+c+d)^2-(ab+ac+ad+bc+bd+cd)$ I do not ...
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### An application of Toeplitz determinant

Given a positive real c between 0 and 2 and d is complex. How to show 4 + Re{(c^2)*d} - |d|^2 - 2(c^2) greater or equal to 0 is equivalent to 2d = c^2 + x((4 - c)^2), for some complex x where x is ...
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### if $a,b\in\mathbb{R}$ proof $|a_{1}|+|a_{2}|\geq |a_{1}-a_{2}|\geq ||a_{1}|-|a_{2}||$ [closed]

$a,b\in\mathbb{R}$ ; $|a_{1}|+|a_{2}|\geq |a_{1}-a_{2}|\geq ||a_{1}|-|a_{2}||$ please prove this inequality system...
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### In a $\triangle ABC,$ If $\cot A+\cot B+\cot C =\sqrt{3},$ Then prove that $\triangle$ is equilateral. [duplicate]

In a $\triangle ABC,$ If $\cot A+\cot B+\cot C =\sqrt{3},$ Then prove that $\triangle$ is equilateral. $\bf{My\; Try::}$ Using Jensen's Inequality, Let $f(x)=\cot x\;,$ Where $x\in (0,\pi),$ ...
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### Sobolev's inequality implies isoperimetric inequality.

How to show that Sobolev's inequality implies isoperimetric inequality in $\Bbb R^d$? I tried to search it on the internet, but most of the results I got are only about the converse direction. So, ...
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### How to find the three largest elements of this matrix?

I have a matrix with positive real numbers $$\mathbf{A}=\begin{pmatrix} a & x & y \\ t & b & z\\ u & v & c\end{pmatrix},$$ where I know that $a,b$ and $c$ are the largest ...
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### Let $a,b$ be real numbers with $|a|<1$. If $2ab+b^2 >3$ then prove that $8a^3-6a+b^3>0$.

Let $a,b$ be real numbers with $|a|<1$. If $2ab+b^2 >3$ then prove that $8a^3-6a+b^3>0$. According to me it is false. For $a=0$ ,$b^2>3$ so $b^3>0$ is false. Am I right?
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### Solve $\sin{2x}-\cos{x}<0$

I'm trying to solve the following inequality: $$\sin{2x}-\cos{x}<0$$ $$2\sin{x}\cos{x}-\cos{x}<0$$ $$\cos{x}(2\sin{x}-1)<0$$ The inequality is verified for ...
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### Showing multiplication inequality using induction

Use induction to show that: $$\frac{1}{2}\frac{3}{4}\dots \frac{2n-1}{2n} < \frac{1}{\sqrt{3n+1}}$$ for $n > 1$.
Currently reading Introduction to Calculus and Analysis by Richard Courant. On page 42 Courant uses the following fact in a larger example explaining why $x^2$ is not uniform continuous (where $a$ and ...