Inequality Definition and 1000 Threads

Is it possible to prove this: x^x + y^y < (x+y)^(x+y) for every x,y >=1 ?

Homework Statement How would Re(z)<0 be graphed? Homework Equations Re(z) is the real part of z The Attempt at a Solution It looks similar to y>x, but only shaded in the third quadrant, how can this be explained? not relevant anymore

Homework Statement Given two arbitrary vectors |\phi_{1}\rangle and |\phi_{2}\rangle belonging to the inner product space \mathcal{H}, the Cauchy-Schwartz inequality states that: |\langle\phi_{1}|\phi_{2}\rangle|^{2} \leq \langle\phi_{1}|\phi_{1}\rangle \langle\phi_{2}|\phi_{2}\rangle...

For $0<x<\dfrac{\pi}{2}$, prove that $\dfrac{\pi^2-x^2}{\pi^2+x^2}>\left(\dfrac{\sin x}{x}\right)^2$. I personally find this challenge very intriguing and I solved it, and feel good about it, hehehe...

So I was helping my sister on homework and there was this problem: 2 abs(2x + 4) +1 > or equal to -3 teacher told her to ignore the -3 and just set it equal to zero. Soo should you? This question got me confused. can't you just go about solving, bringing the 1 to the left and then dividing by 2...

Homework Statement Let ## a,b,c \in \mathbb{R}^{+} ##. Prove that $$ \sqrt[3]{abc} \leq \frac{a+b+c}{3}. $$ Note: ## a,b,c ## can be expressed as ## a = r^3, b = s^3, c = t^3 ## for ## r,s,t > 0##. Homework Equations ## P(a,b,c): a,b,c \in \mathbb{R}^{+} ## ## Q(a,b,c): \sqrt[3]{abc} \leq...

Hi All, Question : Prove that xn<yn , given that x<y and n is odd . Attempt at solution : Assumptions: y-x>0 y2>0 x2>0 So y2x<y3 & x3<x2y So i need to prove that x2y<y2x i.e need to prove then y2x-x2y>0 then yx(y-x)>0, from assumptions y-x>0 so i need to prove that yx>0, so i have 3 cases...

Hi all, I was trying to solve the Inequality x+3x<4 and i found the solution to be x<1, using trial and error. Is there another Logical way or analytic one. Thanks

$x,y,z>1$ please prove : $\dfrac{x^4}{(y-1)^2}+\dfrac{y^4}{(z-1)^2}+\dfrac{z^4}{(x-1)^2}\geq 48$

Homework Statement Homework EquationsThe Attempt at a Solution Hello, I just want to make sure I am doing this right $$<a|b> = a_{x}^{*}b_{x} + a_{y}^{*}b_{y} + a_{z}^{*}b_{z}$$ $$= [(1-i)|x>][-i|x>] + (2 |y>)(-3 |y>) + (0|z>)(|z>)$$ $$=(-i + i^{2})|x> - 6 |y> + 0|z>$$ $$=(-1-i)|x> - 6 |y>...

Hi I'm trying to solve this inequality |1/(2+a)| < 1. 1/(2+a) < 1 ∨ 1/(2+a) > -1 1 < 2+a a > -1 and 1 > -2-a 3 > -a a > -3 I know that the boundaries are -∞ < a < -3 ∨ -1 < a < ∞ What have I done wrong? thanks in advance

Forgive the layman type question but I was doing some reading on Bell's inequality and how it disproves the hidden variable hypothesis in entanglement. The example I looked at was from YouTube I understand the principle of how bell's theorem works and how the tests done on polarisation of...

Hello, all. I am reading Serge Lang's "A First Course in Calculus" in order to get a better understanding of the topic. I thought I would read his review of fundamental concepts, and, naturally, it has been a breeze so far. However, I am stumped when trying to work out this problem: I do not...

Show that the equation $|4x-5|-|3x+1|+|5-x|+|1+x|=0.99$ has no solutions.

In the program below, the result of "difr" is not zero but according to the definition of qx in the second line, I expect it to be zero (because qx^2+ky^2=(\frac{ef-u}{hbarv_f})^2). What is the problem? ef=1;hbarv_f=658;ky=0.0011;u=2.5; qx=sqrt(((ef-u)/hbarv_f)^2-ky.^2)...

Homework Statement [/B] this is the problem , if x and y are real positive numbers , I need to prove $$4x^4 + 4y^3 + 5x^2 + y + 1 \ge 12xy$$ Homework Equations [/B] $$x^2 + y^2 \ge 2xy$$ (Variation of AM GM Theorem) The Attempt at a Solution but $$x^2 + y^2 \ge 2xy $$, so $$6x^2 + 6y^2 \ge...

Prove that for all integers $k>1$: $\left(\dfrac{1+(k+1)^{k+1}}{k+2}\right)^{k-1}>\left(\dfrac{1+k^k}{k+1}\right)^{k}$

Homework Statement Dear Mentors and PF helpers, Here's the question: The roots of a quadratic equation $$3x^2-\sqrt{24}x-2=0$$ are m and n where m > n . Without using a calculator, a) show that $$1/m+1/n=-\sqrt{6}$$ b) find the value of 4/m - 2/n in the form $$\sqrt{a}-\sqrt{b}$$ Homework...

Homework Statement [/B] Prove the following: \sum_{k=1}^{\infty}a_{k}^{2} \leq \left ( \sum_{k=1}^{\infty}a_{k}^{2/3} \right )^{1/2} \left ( \sum_{k=1}^{\infty}a_{k}^{4/3} \right )^{1/2} Homework Equations [/B] The following generalization of Cauchy-Schwarz present in the text (containing...

In Tom Apostol's book "Calculus: Volume 1 (Second Edition) he uses the following inequalities: 0 \lt \cos x \lt \frac{ \sin x }{x} \lt \frac{1}{ \cos x } ... ... ... (1) in order to demonstrate that: \lim_{x \to 0} \frac{ \sin x }{x} = 1... ... BUT ... ... how do we prove (1) ... That is how...

Sketch the region in the plane consisting of all points (x,y) such that lx-yl + lxl - lyl ≤ 2 I don't know exactly the most appreciated solution to this kind of problem. Can you guys show me a clear answer and if possible, a careful graph please?

I am reading the popular-science book A. Zeilinger, Dance of the Photons In the Appendix I have found a surprisingly simple derivation of Bell's inequalities, which, I believe, many people here would like to see. Here it is

Let $x,\,y,\,z$ be positive real numbers such that $xy+yz+zx=3$. Prove the inequality $(x^3-x+5)(y^5-y^3+5)(z^7-z^5+5)\ge 125$.

Homework Statement Dear Mentors and PF helpers, I saw this question on a book but couldn't understand one part of it. Here the question: Solve the following inequality I copied the solution as belowHomework Equations The Attempt at a Solution I don't understand why the numerator in step...

Prove that $\dfrac{\ln x}{x^3-1}<\dfrac{x+1}{3(x^3+x)}$ for $x>0,\,x\ne 1$.

Is here any progress on explaining Bell's Inequality? I do not mean explaining what it is, I mean how it works.

Homework Statement For u and v in R^n prove Minkowski's inequality that \|u + v\| \leq \|u\| + \|v\| using the Cauchy-Schwarz inequality theorem: |u \cdot v| \leq \|u\| \|v\|. Homework Equations Dot product: u \cdot v = u_1 v_1 + u_2 v_2 + \cdots + u_n v_n Norm: \|u \| = \sqrt {u \cdot u}...

Let $a,\,b,\,c,\,d$ be real numbers such that $abcd=1$ and $a+b+c+d>\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}+\dfrac{d}{a}$. Prove that $a+b+c+d<\dfrac{b}{a}+\dfrac{c}{b}+\dfrac{d}{c}+\dfrac{a}{d}$.

I read the following article (I think the author writes on this forum) and thought I understood the reasoning (at least a 0.333 chance of a match, whatever the setting, quantum mechanics for a 120 degree difference in angle predicts a 0.25 chance of a match, measurement shows it to be around...

Homework Statement Here is the problem : https://www.dropbox.com/s/otzzne7wjyuqa5o/question.jpg?dl=0 I've tried solving this inequality but alas,nothing...It's an exam question for my student,and for my great shame,I have no idea how to solve it :(

Hello, In Principles of Mathematical Analysis, the author is attempting to demonstrate that, if ##x > 0## and ##y < z##, then ##xy < xz##, which essentially states that multiplying by a positive number does not disturb the inequality. I am hoping someone will quickly denounce this with an...

Homework Statement Using sandwich theorem evaluvate: $$\lim_{x\rightarrow \infty} \frac{x+7sinx}{-2x+13}$$ Homework Equations Sandwich theorem The Attempt at a Solution ##-7 \leqslant 7sinx \leqslant 7## ##x-7 \leqslant x+7sinx \leqslant x+7## Now my doubt: I want to divide the expression by...

Hallo, could comeone help me to proof this inequality: \frac{m!}{\lambda^{m+1}} \cdot \sum_{j=0}^{m} \frac{\lambda^j}{j!} \leq \frac{2}{\sqrt{\lambda}} . under condition m+1 < \lambda . \lambda is real and m is integer.

Hallo, can someone help me to proof this inequality: (\lambda-(m+1)) \cdot \frac{m!}{\lambda^{m+1}} \cdot \sum_{j=0}^m \frac{\lambda^j}{j!} (m+1-j) \leq \frac{\lambda}{\lambda-m} with condition m+1 < \lambda . \lambda is real und m is integer.

Hallo, can someone help me to proof this inequality: 1-(\lambda-(m+1)) \cdot \frac{m!}{\lambda^{m+1}} \cdot \sum_{j=0}^{m} \frac{\lambda^j}{j!} \leq \frac{\lambda}{(\lambda-(m+1))^2} under condition m+1 < \lambda .

Let $a_i \in \mathbb R^n$ with $a_i = (a_{i}^j)_{j = 1 ... n} = (a_{i}^1, ... ,a_{i}^n)$ for $i = 1, ... , k$ and let $p_1,...,p_k \in \mathbb R_{>1}$ with $\frac1{p_1}+ ... + \frac1{p_k} = 1$ Then show the following inequality by assuming that there are for every $i = 1, ... ,k$ one $N \in...

Hi. After spending an entire day watching videos and reading websites, I'm unable to find the details of a bell's inequality experiment with regards to photons. Could someone please describe such an experiment. What I've learned is that a photon is emitted, say blue. The blue photon is used to...

I went through a paper last week about the Bell inequality and how it is incompatible with QM. Something along the lines of probability in classical regards being 1/3 but in quantum mechanics it is 1/4. It went into some basic principles of how this is determined through quantum entanglement to...

HI all, I have the equation, 6x^2-2>9x for which I'm to find the solution set in interval notation. I've rewritten the inequalty as 6X^2-9x-2=0. I tried to factor, but no go. Then I used the quadratic and got 9+/- rad(129)/-18. The answers I get for x are -1.1309 and .1309. The calculator...

I have read many explanations of Bell’s proof that mention in passing something like “According to QM, the correlation between measurements of spin at different angles should be given by the cosine of the angle between them.” Sometimes they talk about 1-cos(x)/2. Sometimes they talk about...

There are nonnegative real numbers {x}_{1}, {x}_{2}, ... , {x}_{n} such that {x}_{1} + {x}_{2} +...+ {x}_{n} =1 where n \ge 2. Prove that \max\left\{{x}_{1},{x}_{2},...,{x}_{n}\right\} \cdot (1+2 \cdot \sum_{1\le i<j\le n}^{}\min\left\{{x}_{i}, {x}_{j}\right\}) \ge 1 . I noticed that for...

Hi, My first challenge was not very popular so I bring you another one. Let us define f(x)=\dfrac{sin(x)}{x} for x>0. Prove that for every n\in \mathbb{N}, |f^{(n)}(x)|<\dfrac{1}{n+1} where f^{n}(x) denotes the n-th derivative of f

I read that, for ##\delta>0##, if ##\delta<z\leq\pi##, then ##\sin\frac{z}{2}\geq\frac{2\delta}{\pi}##. I cannot prove it. I know that ##\forall x\in\mathbb{R}\quad|\sin x|\leq |x|##, but that does not seem useful here... Thank you so much for any help!

From the differentiation section of my calculus textbook: Sketch the region in the plane consisting of all points $(x,y)$ such that $$2xy\le\left| x-y \right|\le x^2+y^2$$ I have tried to look at the cases: Case 1: $x>y$ $$2xy\le x-y\le x^2+y^2$$ $$0\le (x-y)-2xy\le x^2+y^2-2xy$$ Now I have...

Hi! (Smirk) $$x \in \mathcal{P}A \cup \mathcal{P} B \rightarrow x \in \mathcal{P}A \lor x \in \mathcal{P}B \rightarrow x \subset A \lor x \subset B \rightarrow x \subset A \cup B \rightarrow x \in \mathcal{P} (A \cup B)$$ So, $\mathcal{P}A \cup \mathcal{P}B \subset P(A \cup B) $. The equality...

MIT's Technology Review ran an article on inequality, where they argue that a) it is bad, and b) it is technologically driven, in that it raises some people's income and wealth more than others. I see a tension in these. Suppose I could wave a magic wand, and double the income of everyone...

Question: True or False If x^2<4 then |x|<=2 My solution: I get -2<x<2 when I solve the problem so it should be false. Yet the text says its true? Is this a mistake? If |x| is equal to 2 then it should be a closed interval, not an open interval which seems to be correct to me.

for positive a , b, c prove that $a^4+b^4+c^4 \ge abc(a+b+c)$

Demonstrate that ##|e^{z^2}| \le e^{|z|^2}## We have at our disposal the theorem which states ##Re(z) \le |z|##. Here is my work: ##e^{|z|^2} \ge e^{(Re(z))^2} \iff## By the theorem stated above. ##e^{|z|^2} \ge e^x## We note that ##y^2 \ge 0##, and that multiplying by ##-1## will give us...

Homework Statement If a+b+c=0 then ( (b-c)/a + (c-a)/b + (a-b)/c )( a/(b-c) + b/(c-a) + c/(a-b) ) is equal to: Ans: 9 Homework Equations AM>=GM>=HM Equality holds when all numbers are equal. The Attempt at a Solution I tried using AM>=GM. ( (b-c)/a + (c-a)/b + (a-b)/c + a/(b-c) + b/(c-a) +...