Inequality Definition and 1000 Threads

Find all real $k$ such that $0<k<\pi$ and $\dfrac{8}{3\sin k-\sin 3k}+3\sin^2 k\le 5$.

The cubic polynomial $x^3+mx^2+nx+k=0$ has three distinct real roots but the other polynomial $(x^2+x+2014)^3+m(x^2+x+2014)^2+n(x^2+x+2014)+k=0$ has no real roots. Show that $k+2014n+2014^2m+2014^3>\dfrac{1}{64}$.

The positive real $a$ and $b$ satisfy $b^3+a^2\ge b^4+a^3$. Prove that $b^3+a^3\le 2$.

Hello! (Wasntme) I want to find the Taylor series of the function $f(x)=\log(1+x), x \in (-1,+\infty)$. We take $\xi=0, I=(-1,1)$ It is: $$f'(x)=(1+x)^{-1}, f''(x)=-1 \cdot (1+x)^{-2}, f'''(x)=2 \cdot (1+x)^{-3} , f^{(4)}(x)=-6 \cdot (1+x)^{-4}, f^{(5)}(x)=24(1+x)^{-5}$$ So,we see that...

Hi everyone, I'm a bit confused with this question. An airline demands that all carry-on bags must have length + width + height at most 90cm. What is the maximum volume of a carry-on bag? How do you know this is the maximum? [Note: You can assume that the airline technically mean "all carry...

Could someone please help me with this question: What values are we meant to use to prove this inequality? many thanks

One of the 2 inequalities $(\sin\, x)^{\sin\, x}\,<(\cos\, x)^{\cos\, x} $ and $(\sin\, x)^{\sin\, x}\,>(\cos\, x)^{\cos\, x} $ is always true for all x such that $0 \,< \, x \, < \pi/4$ Identify the inequality and prove it

If $\alpha_n=\arctan n$, prove that $\alpha_{n+1}-\alpha_n<\dfrac{1}{n^2+n}$ for $n=1,\,2,\,\cdots$.

Prove that for any real numbers $a$ and $b$ in $(0,\,1)$, that $(a+b)^{a+b}\le (2a)^a(2b)^b$.

Wonder if this is true or just mistype: |x+y| \leq |x| +|y| If this is true how to proof because cannot find it out anywhere written Regards

Are the less than (<) and greater than(>) relations applicable among complex numbers? By complex numbers I don't mean their modulus, I mean just the raw complex numbers.

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

i am asked to prove phi(t) ≤ phi(s)e^(c(|t-s|)) for t in I. I have to proof this how can I start supposing that t>s and that d/dt(e^(-ct) phi(t))

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?