Inequality Definition and 1000 Threads

Homework Statement Let F and G be bounded functions on S. If f(x) <= g(x) for all x in S prove that inf{f(x):x belongs to S} <= inf{g(x):x belongs to S} Homework Equations None The Attempt at a Solution Basically the idea is to let L0 = inf{f(x):x belongs to S} and L1 = inf{g(x):x...

Homework Statement Show that: (|x+y|)/(1+|x+y|) ≤ ((|x|)/(1+|x|)) + ((|y|)/(1+|y|)) Homework Equations You are given the triangle inequality: |x+y| ≤ |x| + |y| The Attempt at a Solution (This is done from the result, as I haven't been able to find the starting point)...

Homework Statement Describe the region of R^3 that is represented by: Homework Equations x^2 + y^2 + z^2 > 2z The Attempt at a Solution I'm not sure what to do with this at, especially at z=0 and z=2

Homework Statement [PLAIN]http://img703.imageshack.us/img703/7445/unledhpu.png The Attempt at a Solution I am having problems with (c), (e) but I will show yu what I did for the others first. I also I forewarn thee that we haven't learned the Simplex Algorithm yet (we might learn...

Having trouble with this question: The question is: establish the inequality |\inteizzdz| \leq \pi(1-e-R2)/4R on C {z(t) = Reit, t \in [0,\pi/4, R>0 When i saw the modulus of an integral i thought ML inequality. I think the length will be R\pi/4 but I am struggling with...

Prove for all Z E C |ez-1| \leq e|z| - 1 \leq |z|e|z| I think this has to be proven using the triangle inequality but not sure how. Please help. :) thanks

Bell's theorem is generally thought to show that the world cannot be both local and real. http://en.wikipedia.org/wiki/Bell%27s_theorem In simplistic terms, Bell derives an inequality which allegedly must be satisfied if the world is both local and real. In practice, it is found in...

Homework Statement Solve the inequality (2x-3)(4x+5)>(x+6)(x+6) Homework Equations factoring? The Attempt at a Solution I got to the point where (7x)^2-14x-51>0 I can't solve this, because it can't be factored out. So am I doing something wrong?

"Solving Inequality With Complex Numbers" Question Homework Statement What does the inequality pz + conjugate(pz) + c < 0 represent if |p|^2 >c ? Homework Equations p is a constant and a member of the set of complex numbers. c is a constant and a member of the set of real numbers...

Homework Statement Determine the region in the complex plane described by |z-2i| < |z+ i| Homework Equations z= x+ iy |z|= (x2 + y2)1/2 The Attempt at a Solution |z-2i| < |z+ i| |z-2i|/|z+ i| < 1 |z-2i| = [(x-2i)2 + y2]1/2 |z+ i| = [(x+i)2 + y2]1/2 [(x-2i)2 + y2]1/2...

Homework Statement Sketch the graph |Re(z)|>2 Homework Equations z=x+iy The Attempt at a Solution |Re(z)|>2 |Re(x+iy)|>2 |x|>2 |x-0|>2, this is a circle centered at zero with radius 2 4. My question What I'm having a hard time with is the | | notation. Is this the absolute value, or...

Homework Statement f(x) 2 times differentiable function on (0, \infty), and \lim\limits_{x \rightarrow \infty} f(x)=0. there is a M such that M=\sup\limits_{x>0}\vert f^{\prime\prime}(x) \vert. And also for L>0 g(L)=\sup\limits_{x>L}\vert f(x) \vert, and h(L)=\sup\limits_{x>L}\vert...

For my economics/game theory thesis I need to optimize a function subject to an inequality constraint. maximize f(x1, x2) = 1/(x1+x2+y1+y2-w) subject to g(x1, x2) = x1+x2+y1+y2 < w This isn't particularly important, but the x and y variables are quantity of production by a firm. The objective...

Homework Statement Deduce that 0 ≤ ≤ 10/9 for all values of x. Homework Equations The Attempt at a Solution Is it possible to sketch a graph for ? How? Or is there any methods to find the max./min. value of ? Please enlighten me...

Homework Statement This is a question taken from an old exam so I am not sure to which subject in calculus it's connected to... Prove the inequality: \frac{1}{4(ln2)^2}\leq\sum\frac{2^n}{2^(2^n)} (sigma is from 1 to +inf, and the Denominator on the right side is (2^(2^n)) Homework...

Homework Statement Consider any two vectors, |a\rangle and |b\rangle. Prove the Schwartz inequality |\langle a|b \rangle |^2 \leq \langle a|a \rangle \langle b|b \rangle Homework Equations a basic understanding of vector calculus over \mathbb{C}... The Attempt at a Solution I...

Homework Statement Solve \frac{|x^2-5x+4|}{|x^2-4|}\le1 Homework Equations The Attempt at a Solution as |x^2-4|will be positive always cross multiply and take 1 to other side of equation solve by taking LCM we get |x^2-5x+4|-(x^2-4)\le0 on solving we get...

Homework Statement Solve the inequation (x^2+3x+1)(x^2+3x-3)\ge 5The Attempt at a Solution on opening the brackets i got x^2(x^2+3x-3)+3x(x^2+3x-3)+1(x^2+3x-3)\ge 5 x^4+3x^3-3x^2+3x^3+9x^2-9x+x^2+3x-3-5\ge 0 x^4+6x^3+7x^2-6x-8\ge 0 am i right?? after that what should i do??

Solve for x-- the inequality of quadratic Homework Statement Solve \frac{2x}{x^2-9}\le\frac{1}{x+2} The Attempt at a Solution x^2-9\not=0 .'. x\in R-\{-3,3\} and x+2\not=0 .'. x\in R-\{-2\} then converting the original inequality to (2x)(x+2)\le(x^2-9)...

Homework Statement ∫(from pi/4 to pi/2)sin x/x ≤ 1/√2. Homework Equations The Attempt at a Solution I know the pi/4≤x≤pi/2 and so 1/√2 ≤ sin x ≤ 1 and i have tried to manipulate this to no end and it has annoyed the living daylights out of me

Homework Statement Part of an \epsilon-\delta proof about whether or not f + 2g is continuous at x = a provided that f and g are continuous at x = a The Attempt at a Solution I've got the proof (I hope), but I'm uncertain about whether I can do the following...

Homework Statement Prove that if x < y, and n is odd, then x^{n}< y^{n} The Attempt at a Solution My attempt was to solve three different cases: Case 1: If 0 \leq x < y, we have y-x > 0 y*y*...*y > 0 (closure of the positive numbers under multiplication)...

Homework Statement Solve for x Homework Equations x^3 < x The Attempt at a Solution x^3 < x x^3 - x < 0 x(x^2 - 1) < 0 x(x+1)(x-1) < 0 For the expression on the left to be less than zero, it has to be two positives + negative or three negatives right? I've tried setting...

I saw this rather odd symbol of the the greater sign on top of the less sign in my lecture notes. I am wondering if there is a name for this symbol and if signifies 'equal to' as well?

Hello I need to prove this inequality: http://img6.imageshack.us/img6/2047/unledwp.jpg Uploaded with ImageShack.us where y=im(z) ,x=Re(z). I used the triangle inequality but I got stuck. Can someone show me how to do it? specially the left side of the inequality. thanks

Can't figure this out and hope to get some help, TIA! a,b,c >= 0 and a+b+c=3 Prove that a²+b²+c²+ab+bc+ca >= 6

I don't understand how it is possible to show using the Minkowski's Inequality that (\sum x_i )^a \leq \sum x_i^a where x_i \geq 0 \forall i and 0<a<1 . I also tried to prove this without using Minkowski, but to no avail. This is driving me crazy although it seems to be trivial in...

Homework Statement I'm actually only concerned here with proving equality. I would like some review of my proof before I crawl back to my professor again with what I think is a valid proof. The Attempt at a Solution Show: \frac{x_1+x_2+...+x_n}{n}=\sqrt[n]{x_1x_2\cdots x_n} \Leftrightarrow...

Homework Statement Suppose that a and b are nonzero real numbers. Prove that if a<1/a<b<1/b then a<-1.The Attempt at a Solution So after a while I realized that I could prove that a<-1 by contradiction but first I have to prove that a<0. I figured out how to prove it but I'm not sure if my...

Hi, Quick question here: I know that C-S inequality in general states that |<x,y>| \leq \sqrt{<x,x>} \cdot \sqrt{<y,y>} and, in the case of L^2(a,b)functions (or L^2(R) functions, for that matter), this translates to |\int^{b}_{a}f(x)g(x)dx| \leq \sqrt{\int^{b}_{a}|f(x)|^2dx} \cdot...

Homework Statement prove the inequality. Homework Equations (in the attached file.) The Attempt at a Solution The derivative of (1/2)^m where m=2^n is exactly what I need. But I can’t find the sum of (1/2)^m (because as it’s not geometric series, m doesn’t run from 0 to inf). by...

Prove that R(p,q) \leq \left(\stackrel{p+q-2}{p-1}\right) where p and q are positive integers I'm supposed to use induction on the inequality R(p,q) \leq R(p-1,q) + R(p,q-1) , but I'm having difficulty there. How do I go about doing this? I can show it's true for p=q=1. But, I can't...

I don't seem to understand Clausius inequality at all. Really. It was deduced to me that the Clausius inequality is given by dS = \frac{\delta Q_i}{T} > 0 where Q_i is the irreversible heat transferred to a system. Though I cannot find a way to prove an assertion my teacher said: through...

attached are the problems (actually i don't think i bothered with #96) I'm having trouble with. attached is ONE of my attempts and attached is the book's answers. I have NO idea where to even begin with these.

Homework Statement Consider \sum_{1}^{\infty} a_{n}, a_{n} \neq 0 Show that \underline{\lim\limits_{n \rightarrow \infty}}|\frac{a_{n+1}}{a_{n}}| \leq \underline{\lim\limits_{n \rightarrow \infty}}\sqrt[n]{|a_{n}|}\leq \overline{\lim\limits_{n \rightarrow \infty}}\sqrt[n]{|a_{n}|)}...

Im curios as to why the inquality is ||x+y||\leq||X||+||y|| but the end of the proof is =(||x||+||Y||)^2 where does the less than symbol disappear too

I know that if \forall x \in E \subset \mathbb{R}^n we have f(x) \le g(x) then it is true that \int_E f \le \int_E g . However, is it also true that if \forall x \in E we have f(x) < g(x) then \int_E f < \int_E g ?

Hi guys, I am reading a proof on Holder's inequality. There is a line I don't understand. Here is the extract from Kolmogorov & Fomin, Introductory Real Analysis. "The proof of [Minkowski's inequality] is in turn based on Holder's inequality \sum_{k=1}^n |a_k b_k|\leq...

Homework Statement (3x+1)/(x+4)>=1 Homework Equations The Attempt at a Solution (3x+1)>=(x+4) 2x>=3 x>=3/2 But this is wrong?? Why?

Homework Statement i got stuck at the question below:- Homework Equations The Attempt at a Solution I tried to solve it by simplifying it but i got stuck at:- Please help.

Homework Statement I'm reading the proof for the reverse triangle inequality, but I don't understand what is meant by "by symmetry" Homework Equations The Attempt at a Solution (X,d) is a metric space prove: |d(x,y) - d(x,z)| <= d(z,y) The triangle inequality d(x,y) <=...

Homework Statement Prove that if |x-x_0| < \textrm{min} \bigg ( \frac{\epsilon}{2|y_0|+1},1 \bigg ) and |y-y_0| < \frac{\epsilon}{2|x_0|+1} then xy-x_0y_0<\epsilon Homework Equations We can use basic algebra and the following axioms: For any number a, one and only one of the following...

Homework Statement For a symmetric matrix A, use the notation \lambda_{k}\left(A\right) to denote the k^{th} largest eigenvalue, thus \lambda_{n}\left(A\righ)<=...<=\lambda_{2}\left(A\right)<=\lambda_{1}\left(A\right) Now suppose A and A+E are nxn symmetric matrices, prove the following...

Assume: p>1, x>0, y>0 a \geq 1 \geq b > 0 \frac{a^2}{p^2}+(1-\frac{1}{p^2})b^2 \leq 1 \frac{x^2}{a^2}+\frac{y^2}{b^2} \leq 1 Prove: \frac{x}{p}+y\sqrt{1-\frac{1}{p^2}} \leq 1 I've been trying for 3 days and it's driving me crazy. Any ideas?

Revenue Equation: R(x)=-x^2+10x Cost Equation: C(x)= 4X+5 Average profit= profit equation, P(x)/x therefore p(x)= R(x)-C(x)=-x^2+6x-5 (-x^2+6x-5)/x=(-1(x-5)(x-1))/x, I then found that x is positive between 1 and 5, therefore average profit is positive in that range, however, the answer...

Homework Statement Prove that (n + 1)n - 1 < nn for n ∈ Z+. [Hint: Induction is suggested. Write out the induction statement explicitly. Make one side of the inequality look like your induction hypothesis.] Homework Equations The Attempt at a Solution ^ That's what I have so far. I'm good...

How can I solve, step by step, this inequality ? The result I have is [ 1 , (-1 + sqrt33)/2 ] but the result should be [ 0 , (-1 + sqrt33)/2 ]|x+2|-|x-1|\geq\sqrt{x^2+x+1} thanks for ur help =)

Homework Statement I have to approximate sin(1/2) with the taylor inequality Homework Equations taylors inequality |Rn(x)| ≤ M/(n+1)! | x-a|n+1 The Attempt at a Solution Im not really sure what the significance of this is, but ill do the derivatives f(x) = sin(x) f'(x) = cos(x) f''(x) =...

Homework Statement Let\; S = \sqrt(1) + \sqrt(2) + \sqrt(3) + ... + \sqrt(10000) \;and\; I = \int_{0}^{10000} \sqrt x dx[/itex] Show that I \leq S \leq I+100 The Attempt at a Solution Consider\; I\;=\;\int_{0}^{10000} \sqrt x dx I\;=\; \frac{(10000)^{3/2}}{(3/2)}...

Homework Statement the actual problem is to show that d(x,y)=d1(x,y)/[1+d1(x,y)] expresses a distance in R^n if d1(x,y) is a distance in R^n.Based on theory I have to show that i) d(x,y)>=0 , ii)d(x,y)=d(y,x) and iii)d(x,y)<= d(x,z)+d(z,y) i've proven the first two so basically how can i...