Inequality — 400 discussions

$x$ and $y$ are two real numbers such that $x^3-y^3=2$ and $x^5-y^5\ge 4$. Prove that $x^2+y^2\gt 2$.

My previous thread on this topic got a bit messy as the gist of the argument was in the middle of the thread and turned out wrong. Hence this new updated version. One of my favourite articles on Bell's Theorem can be found at...

Prove $$\frac{ab}{a+b+ab}+\frac{bc}{b+c+bc}+\frac{ca}{c+a+ca}\le \frac{a^2+b^2+c^2+6}{9}$$ for all positive real $a,\,b$ and $c$ and $a,\,b,\,c\ne 0$.

I've seen some articles using particle spin experiments to 'prove' that the results violate Bell's inequality and consequently local reality. I've also seen stated that the same experiments can be done using other particle attributes such as polarisation. I can see how with polarisation, you...

Homework Statement ##|4^{3x}-2^{4x+2}*3^{x+1}+20*12^x*3^x| > 8*6^x(8^{x-1}+6^x)## For some numbers ##a, b, c, d## such that ##-\infty < a <b < c <d < +\infty ## the real solution set to the given inequality is of the form ##(-\infty, a] \cup [b, c] \cup [d, +\infty)## Prove it by arriving at...

Prove that $$\frac{\sqrt{a^2+b^2}}{a+b}+\sqrt{\frac{ab}{a^2+b^2}}\le \sqrt{2}$$ for all positive reals $a$ and $b$.

Hello! Say we have an inequality that says that ##f(x, y)>c## where ##f(x, y)## is a function of two variables and ##c## is a constant. Assume that we know this inequality to be true when ##x=a## and ##y=b##. If you show that the partial derivatives of ##f(x, y)## with respect to ##x## and ##y##...

If $\left| a \right| \le b$, then $-b\le a\le b$. Let $a,b \in\Bbb{R}$ The definition of the absolute value is $ \left| x \right|= x, x\ge 0$ and $\left| x \right|=-x, x< 0$, where x is some real number. Case I:$a\ge 0$, $\left| a \right|=a>b$ Case II: a<0, $\left| a \right|=-a<b$the solution...

I am not sure if I am allowed to ask this, but here's my shot: I find all the explanations of Bell's theorem summed up here, very different in interpretation and also (for me) incomprehensible. I have these simple questions: How does the Bell inequality, stated as N(A, not B) + N(B, not C) ≥...

How can we prove $$n^n\cdot \left(\frac{n+1}{2}\right)^{2n}\geq \left(\frac{n+1}{2}\right)^3$$ I did not understand from where i have start.

Based on Schwartz inequality, I am trying to figure out why there should/can be the "s" variable which is the lower bound of the integration in the RHS of the following inequality: ## \left \|\int_{-s}^{0} A(t+r)Z(t+r) dr \right \|^{2} \leq s\int_{-s}^{0}\left \| A(t+r)Z(t+r) \right \|^{2} dr...

Let $a,\,b,\,c$ be real numbers such that $a\ge b\ge c>0$. Prove that $$\frac{a^2-b^2}{c}+\frac{c^2-b^2}{a}+\frac{a^2-c^2}{b}\ge 3a-4b+c$$.

For positive real numbers $a,\,b,\,c$, prove the inequality: $$a + b + c ≥ \frac{a(b + 1)}{a + 1} + \frac{b(c + 1)}{b + 1}+ \frac{c(a + 1)}{c + 1}$$

If $a$ and $b$ are two positive real, and that $a^3+b^3=a-b$, prove that $2\left(\sqrt{2}-1\right)a^2-b^2\lt 1$.

[b[1. Homework Statement [/b] ##|4^{3x}-2^{4x+2}*3^{x+1}+20*12^x*3^x| \ge 8*6^x(8^{x-1}+6^x)## The sets containing the real solutions for some numbers ##a, b, c, d,## such that ##-\infty < a < b < c < d < +\infty## is of the form ##(-\infty, a] \cup [b, c] \cup [d, +\infty)##. Prove it by...

Let a>0. It is also true that a+b>0. Can we prove that b>0 always? My attempt b>-a... but 0>-a. therefore min (b,0)>-a case 1: b <0. If, b <-a, then a <-b so a+b <b-b so, a+b <0. Contradiction Hence b>0. Is my proof right?

if x+y ≥ 2 it contains all the point in the line x+y =2 and the half plane above it. but ,graph if x-y ≥ 2 then if consider a line x-y= 2 the inequality represents the line and the half plane below it . i don't understand why it represents the half line below it why not above ?

Homework Statement Let $$(X(n), n ∈ [1, 2])$$ be a stationary zero-mean Gaussian process with autocorrelation function $$R_X(0) = 1; R_X(+-1) = \rho$$ for a constant ρ ∈ [−1, 1]. Show that for each x ∈ R it holds that $$max_{n∈[1,2]} P(X(n) > x) ≤ P (max_{n∈[1,2]} X(n) > x)$$ Are there any...

Prove that $\sqrt{8}^{\sqrt{7}}<\sqrt{7}^{\sqrt{8}}$.

I would like to show you how to use Schwarz inequality to prove some important general theorems and solve problems about vectors in Minkowski spacetime. Okay, Schwarz inequality states that \left| U^{k}V^{k}\right| \leq \sqrt{(U^{i})^{2}(V^{j})^{2}}. \ \ i,j,k =1,2,3 \ \ \ (1) And, the...

I am reading the proof of the Riemannian Penrose Inequality (http://en.wikipedia.org/wiki/Riemannian_Penrose_inequality) by Huisken and Ilmamen in "The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality" and I was wondering why they restrict their proof to the dimension ##n=3##...

Why would an inequality sign flip in an answer. For example: 16 < -s -6 The answer is given s < -22 I had -s > 22 I am thinking it is because when you x by -1 to keep from having a negative variable the inequality sign flips...is this why?

$m,n,k\in N$, and $m>1,n>1$ prove : $(3^{m+1}-1)\times (5^{n+1}-1)\times(7^{k+1}-1)>98\times 3^m\times 5^n\times7^k$

I was wondering if someone could explain to me how John Bell and Anton Zeilinger have attempted to prove there is no definite reality in the sub-atomic world. How could it ever be proven there are no hidden variables that humans just don't or can't know about? If I am not mistaken, Einstein for...

There are several Bell inequalities involving 4 values (e.g. CHSH where they are sometimes denoted by Q, R, S, T). The original Bell inequality involved 6. All being refuted by QM. Is it known whether there is one with only 3 values? I can prove there isn't one with 2 values.

Prove for all positive real $a,\,b,\,c,\,x,\,y,\,z$ that $\dfrac{a^3}{x}+\dfrac{b^3}{y}+\dfrac{c^3}{z}\ge \dfrac{(a+b+c)^3}{3(x+y+z)}$.

Does anyone know a good resource for exercises on these topics?

Prove the following: $$\sum_{k=n}^{2n-2}\frac{1}{k^2}<\frac{1}{2n-1}$$ where $$2\le n$$

Let $a,\,b$ and $c$ be positive real numbers satisfying $a+b+c=1$. Prove that $9abc\ge7(ab+bc+ca)-2$.

Homework Statement How many non-negative integer solutions are there to the equation x1 + x2 + x3 + x4 + x5 < 11, (i)if there are no restrictions? (ii)How many solutions are there if x1 > 3? (iii)How many solutions are there if each xi < 3? Homework Equations N/A The Attempt at a Solution...

Homework Statement Homework Equations With the regards to posting such a incomplete equation, I will soon put in the updated one Thank you The Attempt at a Solution visual graph... didn't help

For the positive real numbers $x,\,y$ and $z$ that satisfy $\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=3$, prove that $\dfrac{1}{\sqrt{x^3+1}}+\dfrac{1}{\sqrt{y^3+1}}+\dfrac{1}{\sqrt{z^3+1}}\le \dfrac{3}{\sqrt{2}}$.

[Mentor's note: Moved from a thread about field theories as this is just about the basic meaning of the theorem for ordinary entangled particles]] Suppose that there are photon spin outcomes that are pre existing from entanglement or from local hidden variables. For every detector angle...

Homework Statement The number of customers visiting a store during a day is a random variable with mean EX=100and variance Var(X)=225. Using Chebyshev's inequality, find an upper bound for having more than 120 or less than 80customers in a day. That is, find an upper bound on P(X≤80 or X≥120)...

Homework Statement Let X∼Geometric(p). Using Markov's inequality find an upper bound for P(X≥a), for a positive integer a. Compare the upper bound with the real value of P(X≥a). Then, using Chebyshev's inequality, find an upper bound for P(|X - EX| ≥ b). Homework Equations P(X≥a) ≤ Ex / a...

Let $a\in \Bbb{Z^+}$, prove that $\dfrac{2}{2-\sqrt{2}}>\dfrac{1}{1\sqrt{1}}+\dfrac{1}{2\sqrt{2}}+\dfrac{1}{3\sqrt{3}}+\cdots+\dfrac{1}{a\sqrt{a}}$.

Is this true? (∞ - 1) < ∞

I'm not really sure if this is true, which is why I want your opinion. I have been trying to prove it, but it will help me a lot if someone can confirm this. Let ## v_{1}, v_{2} ... v_{n} ## be vectors in a complex inner product space ##V##. Suppose that ## | v_{1} + v_{2} +...+ v_{n}| =...

Homework Statement Prove that if ## |x-x_0|<\min (\frac {\epsilon}{2(|y_0|+1)},1)## and ##|y-y_0|<\frac{\epsilon}{2(|x_0|+1)} ## then ## |xy-x_0y_0|<\epsilon ## Homework Equations N/A The Attempt at a Solution From the first inequality I can see that: ## |x-x_0|<\frac...

Hello, I'm having trouble with an assigned problem, not really sure where to begin with it: Prove that if $$a \in R$$ and $$b \in R$$ such that $$0 < b < a$$, then $${a}^{n} - {b}^{n} \le {na}^{n-1}(a-b)$$, where n is a positive integer, using a direct proof. Pointers or the whole proof would...

Homework Statement Using the principle of mathematical induction, prove that for all n>=10, 2^n>n^3 Homework Equations 2^(n+1) = 2(2^n) (n+1)^3 = n^3 + 3n^2 + 3n +1 The Attempt at a Solution i) (Base case) Statement is true for n=10 ii)(inductive step) Suppose 2^n > n^3 for some integer >=...

The Cauchy-Schwartz inequality (\sum_{i=1}^n x_i^2)(\sum_{i=1}^n y_i^2) - (\sum_{i=1}^n x_iy_i)^2 \geq 0 holds with equality (or is as "small" as possible) if there exists an a \gt 0 such that x_i=ay_i for all i=1,...,n . But when is the inequality as "large" as possible? That is, can we...

I need help determining the interval notation of the inequality below: -9<1/x<=1

Solve the following inequality. Represent your answer graphically: ## |z-1| + |z-5| < 4 ## Homework Equations ## z = a + bi \\ |x+y| \leq |x| + |y| ## Triangle inequality The Attempt at a Solution ## |z-1| + |z-5| < 4 \\ \\ x = z-1 \ \ , \ \ y = z-5 \\ \\ |z-1+z-5| \leq |z-1| + |z-5| \\...

Let ${y}_{n}$ be a arbitrary sequence in X metric space and ${y}_{m+1}$ convergent to ${x}^{*}$ in X...İn this case by using triangle inequality can we say that ${y}_{n}\to {x}^{*}$

Can somebody help me please, I've tried solving this for hours but I still couldn't get it. Given that a, b, c, d are positive integers and a+b=c+d. Prove that if a∗b < c∗d, then a∗log(a)+b∗log(b) > c∗log(c)+d∗log(d) How do I do it?

Can someone explain to be in detail what is quadratic inequality? It's rather confusing. Thank you

Given $x,\,y,\,z$ are positive real numbers. Prove that $\dfrac{xy}{x^2+xy+y^2}-\dfrac{1}{9}+\dfrac{yz}{y^2+yz+z^2}-\dfrac{1}{9}+\dfrac{zx}{z^2+zx+x^2}-\dfrac{1}{9}\le \dfrac{2\sqrt{xy+yz+zx}}{3\sqrt{x^2+y^2+z^2}}$

Prove $\sqrt[3]{1−12\sqrt[3]{65^2} + 48\sqrt[3]{65}} -\sqrt[3]{63}\gt \sqrt[3]{1−48\sqrt[3]{63} + 36\sqrt[3]{147}} -4 $

Hello all, I want to prove the following inequality. sin(x)<x for all x>0. Now I figured that I put a function f(x)=x-sin(x), and show that it is increasing for all x>0. But this alone doesn't prove it. I need to show we have inequality from the start. I can't show that lim f(x) as x->0 is...