Inequality Definition and 1000 Threads

Problem: If A is the area and 2s the sum of three sides of a triangle, then: A)$A\leq \frac{s^2}{3\sqrt{3}}$ B)$A=\frac{s^2}{2}$ C)$A>\frac{s^2}{\sqrt{3}}$ D)None Attempt: From heron's formula: $$A=\sqrt{s(s-a)(s-b)(s-c)}$$ From AM-GM: $$\frac{s+(s-a)+(s-b)+(s-c)}{4}\geq...

Here's the claim: Assume that A and B are both symmetric matrices of the same size. Also assume that at least other one of them does not have negative eigenvalues. Then \textrm{Tr}(ABAB)\geq 0 I don't know how to prove this!

Prove that $\dfrac{1}{\sqrt{4x}}\le\left( \dfrac{1}{2} \right)\left( \dfrac{3}{4} \right)\cdots\left( \dfrac{2x-1}{2x} \right)<\dfrac{1}{\sqrt{2x}}$.

Show that $e^\dfrac{1}{e}_{\phantom{i}}+e^{\dfrac{1}{\pi}}_{\phantom{i}} \ge2e^{\dfrac{1}{3}}_{\phantom{i}}$.

Let $(a,b,c)$ be a Pythagorean triple, specifically, a triplet of positive integers with property $a^2 + b^2 = c^2$. Show that $(\frac ca + \frac cb)^2 > 8$. EDIT: Added a small clarification.

Let $x,y,z$ be positive real numbers. Show that $x^2 + xy^2 + xyz^2 \geq 4xyz - 4$.

Let $x,\,y$ be positive integers such that $\dfrac{7}{10}<\dfrac{x}{y}<\dfrac{11}{15}$. Find the smallest possible value of $y$.

If 0 ≤ a < b and 0 ≤ c <d, then prove that ac < bd I have taken the proof approach from some previous problems in Spivak's book on Calculus (3rd edition). This is problem 5.(viii) in chapter 1: Basic Properties of Numbers. I did as follows: If a = 0 or c = 0, then ac = 0, but since...

(BMO, 2013) The angles $A$, $B$, $C$ of a triangle are measures in degrees, and the lengths of the opposite sides are $a$,$b$,$c$ respectively. Prove: \[ 60^\circ \leq \frac{aA + bB + cC}{a + b + c} < 90^\circ. \] Edit: Update to include the degree symbol for clarification. Thanks, anemone.

Show that $\dfrac{1}{2} \cdot \dfrac{3}{4} \cdot \dfrac{5}{6} \cdots \dfrac{1997}{1998} >\dfrac{1}{1999}$, where the use of induction method is not allowed.

Homework Statement Homework Equations P (|Y - μ| < kσ) ≥ 1 - Var(Y)/(k2σ2) = 1 - 1/k2 ?? The Attempt at a Solution using the equation above 1 - 1/k2 = .9 .1 = 1/k2 k2 = 10 k = √10 = 3.162 k = number of standard deviations. After this I don't know where to go...

Homework Statement Let ##f:[a,b]\rightarrow\mathbb{R}## and ##g:[a,b]\rightarrow\mathbb{R}## be continuous functions having the property ##f(x)\leq g(x)## for all ##x\in[a,b]##. Prove ##\int_a^b \mathrm f <\int_a^b\mathrm g## iff there exists a point ##x_0## in ##[a,b]## at which...

$0<\alpha < \dfrac {\pi}{2}$ $0<\beta < \dfrac {\pi}{2}$ prove: $(1): \,\, \dfrac{1}{ \cos^2 \alpha}+ \dfrac {1}{ \sin^2 \alpha \, \sin^2 \beta \, \cos^2 \beta} \geq 9 $ determine the values of $ \tan \alpha$ and $ \tan \beta $ when : $(2): \: \dfrac{1}{ \cos^2 \alpha}+ \dfrac {1}{ \sin^2 \alpha...

$a,b,c\geq 1$ prove :$\dfrac {ab+c}{c+1}+\dfrac {bc+a}{a+1}+\dfrac {ca+b}{b+1}\geq\dfrac {18} {a+b+c+3}$

Homework Statement I'm working on some MIT OCW (http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-436j-fundamentals-of-probability-fall-2008/assignments/MIT6_436JF08_hw01.pdf). I've attempted problem #5, just looking for some comments on the quality / validity of my...

Homework Statement Solve the given inequality by interpreting it as a statement about distances in the real line: |x-3| < 2|x| Homework Equations The Attempt at a Solution I have no clue what to do here and I do not understand the answer in the textbook Goes something like...

Why correct this inequality, log(d/(d-1))>1/d for d≥2?

Hello, I am given that β > α, which can be written as β - α > 0. What justification would I have to use in order to conclude that α - 2β < 0, given that the preceding propositions are true? Could someone possibly help me?

Introduction to Operator Theory and Invariant Subspaces - B. Beauzamy - Google BooksIn page 144 of this preview I don't know how they obtain the inequality in (1). It looked like cauchy schwarz but I don't think it is. I also don't know how they connect the norm of the integral to the supremum...

Let $a$ and $b$ be positive integers. Show that $\dfrac{(a+b)!}{(a+b)^{a+b}}\le \dfrac{a! \cdot b!}{a^ab^b}$.

Homework Statement Homework Equations I have to use these set identities: The Attempt at a Solution Pretty sure this is impossible since it's an inequality.

Show that $\dfrac{1}{2}\cdot\dfrac{3}{4}\cdot\dfrac{5}{6} \cdots\dfrac{999999}{1000000}<\dfrac{1}{1000}$

Prove that if x and y are not both , then x^4+x^3y+x^2y^2+xy^3+y^4 > 0 I have no idea how to start this proof, can anyone give me an idea?

Homework Statement Solve the inequality for x, given that (4x - 16) / [(x - 3)(x - 9)] < 0 Homework Equations I can't think of any for this type of problem... The Attempt at a Solution (4x - 16) / [(x - 3)(x - 9)] < 0 4(x - 4) / [(x - 3)(x - 9)] < 0 I'm not sure where...

If $a,\,b,\,c$ are positive numbers, show that $8(a^3+b^3+c^3)\ge (a+b)^3+(a+c)^3+(b+c)^3$.

Homework Statement Show that the inequality\left|\frac{z^2-2z+4}{3x+10}\right|\leq3holds for all z\in\mathbb{C} such that |z|=2 Homework Equations Triangle inequality The Attempt at a Solution I'm not really sure how to go about this. the x is throwing me off. Should I write it out with...

Hi I have a problem about Inequality. Let's suppose we have a inequality like this: axb≥d and cxd≥d so I want to find connection between a and d its possible to do something like that Thanks

1. |(x2+2x+1)/(x2+3)|≤ M. Find the value of M when |x|≤ 3. 2. |u+v|=|u|+|v| 3. I understand that you start off by distributing the absolute value symbols into the individual terms as above. Then you maximize the numerator, using 3 as the value for x. However, my professor then...

Dear all, I've encountered some problems while looking through the book called "Operator Algebras" by Bruce Blackadar. At the very beginning there is a definition of pre-inner product on the complex vector space: briefly, it's the same as the inner product, but the necessity of x=0 when [x,x]=0...

Homework Statement I'm trying to show that if ##a \approx 1##, then $$-1 \leq \frac{1-a}{a} \leq 1$$ I've started off trying a contradiction, i.e. suppose $$ \frac{|1-a|}{a} > 1$$ either i) $$\frac{1-a}{a} < -1$$ then multiply by a and add a to show $$1 < 0$$ which is clearly...

How can it be shown that $$16x\cos(8x)+4x\sin(8x)-2\sin(8x)<|17x|?$$ This problem arises from work with damped motion in spring-mass systems in Differential Equations. I have gotten to this inequality after some algebraic manipulation, but am completely stuck here. Here is the illustrative...

So hi, there's one little thing which I'm not understanding in the proof. After the inequality Spivak considers the two expressions to be equal. Why?!? I just don't see why we can't continue with the inequality and when we have factorized the identity to (|a|+|b|)^2 we can just replace...

Prove that $\sqrt[3]{\dfrac{2}{1}}+\sqrt[3]{\dfrac{3}{2}}+\cdots+\sqrt[3]{\dfrac{996}{995}}-\dfrac{1989}{2}<\dfrac{1}{3}+\dfrac{1}{6}+\cdots+ \dfrac{1}{8961}$

Prove that $\dfrac{1}{\dfrac{1}{a}+\dfrac{1}{b}}+\dfrac{1}{ \dfrac{1}{c}+\dfrac{1}{d}} \le \dfrac{1}{\dfrac{1}{a+c}+\dfrac{1}{b+d}}$ for all positive real numbers $a, b, c, d$.

Hi, With the following norm inequality: ||Av|| ≤ ||A||||v|| implies ||A|| = supv [ ||Av||/||v|| ] I understand that sup is the upper bound of a set B, or least upper bound if B is a subset of A, where the upper bounds are elements of both B and A. Is this saying that the norm of A...

Let a, b, and c be positive integers such that: \frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{c^2} Find the sum of all possible values of a that are less than or equal to 100. My shot at it: I think may be that equation would also equals to a^2 + b^2 = c^2... So may be a is the sum of all the...

Triangle ABC with side lengths a,b,c please prove : $ \sqrt {ab}+\sqrt {bc}+\sqrt {ca}\leq a+b+c<2\sqrt {ab}+2\sqrt {bc}+2\sqrt {ca}$

The Born in ineqality fails experimentally and fails to make sense if we think of quanta as particles. Does it make sense otherwise?

Homework Statement Show that if 0 \le x \le a, and n is a natural number, then 1+\frac{x}{1!}+\frac{x^2}{2!}+...+\frac{x^n}{n!} \le e^x \le 1+\frac{x}{1!}+\frac{x^2}{2!}+...+\frac{x^n}{n!}+\frac{e^ax^{n+1}}{(n+1)!} Homework Equations I used Taylor's theorem to prove e^x is equal to the LHS...

For any triangle $ABC$, prove that $\cos \dfrac{A}{2} \cot \dfrac{A}{2}+\cos \dfrac{B}{2} \cot \dfrac{B}{2}+\cos \dfrac{C}{2} \cot \dfrac{C}{2} \ge \dfrac{\sqrt{3}}{2} \left( \cot \dfrac{A}{2}+\cot \dfrac{B}{2}+\cot \dfrac{C}{2} \right)$

Problem: Prove: $$\sqrt{C_1}+\sqrt{C_2}+\sqrt{C_3}+...+\sqrt{C_n} \leq 2^{n-1}+\frac{n-1}{2}$$ where $C_0,C_1,C_2,...,C_n$ are combinatorial coefficients in the expansion of $(1+x)^n$, $n \in \mathbb{N}$. Attempt: I thought of using the RMS-AM inequality and got...

Let $x, y$ be real numbers such that $9x^2+8xy+7y^2 \le 6$. Show that $7x+12xy+5y \le 9$.

Homework Statement Let 0 ≤ x_1 ≤ x_2 ≤ ... ≤ x_n and x_1 + x_2 + ... + x_n = 1 , All the 'x' are real and n is a natural number. Prove the following: (1+x_1^21^2)(1+x_2^22^2)...(1 + x_n^2n^2) ≥ \frac{2n^2+9n+1}{6n} Homework Equations The Attempt at a Solution...

I'm stuck on one aspect of the proof on page 105 of the 2nd edition. Equation 6.13 is necessary for the inequality to be an equality as it says but they never seem to account for inequality 6.11. Specifically, I don't see how this satisfies 2 Re<u,v> = 2 |<u,v>| Thanks for any guidance.

Here's a claim: Assume that a function f:[a,b]\to\mathbb{R} is differentiable at all points in its domain. Then the inequality |f(b) - f(a)| \leq \int\limits_{[a,b]}|f'(x)|dm(x) holds. The integral is the Lebesgue integral. Looks simple, but I don't know if this is true. There exists...

Are there any modern interpretations of QM that predict the correlations in a Bell Inequality violation ? Preferably a local non realistic model based on mechanisms.

Homework Statement I am asked to discover and prove the inequality relationship between root mean square, arithmetic mean, geometric mean, and harmonic mean. Homework Equations Let a,b, be non-negative integers. (a-b)2 ≥ 0 and (√a-√b)2 ≥ 0 The Attempt at a Solution Using (a-b)2 ≥...

Leaving aside the debatable point about whether bell's violation rules out all local theories or just local realism, is there general agreement that if the assumptions are valid, violation of Leggett's inequalities rules out any non-local model that treats properties other than position as real...

[solved]Complex inequality Homework Statement You have the two inequalities, where k is a complex number; |k+\sqrt{k^2-1}|<1 and |k-\sqrt{k^2-1}| <1 Show that if ##|k|>1##, then the second inequality is fulfilled, while the first one is impossible for any value of k. The Attempt...

Homework Statement Question: Use induction to prove that 3^n > n x 2^n for every natural number n ≥ 3 Homework Equations N/A The Attempt at a Solution Answer: Step 1: 3^3 > 3 x 2^3 ⇒ 27 > 24 Step 2: Assume 3^k > k x 2^k Step 3: 3^(k+1) > (k+1) x 2^(k+1) ⇒ 3 x 3^k > k x 2^(k+1) +...