Inequality Definition and 1000 Threads

I posted the paragraph with the inequality where I am confused... I also added a portion i highlighted in red

Homework Statement Use the Cauchy-Schwarz inequality to prove that for all real values of a, b, and theta (which ill denote as θ), (a cosθ + b sinθ)2 ≤ a2 + b2 Homework Equations so the Cauchy-Schwarz inequality is | < u,v>| ≤ ||u|| ||v|| The Attempt at a Solution I'm having...

Homework Statement Let u = [a b] and v = [1 1]. Use the Cauchy-Schwarz inequality to show that (a+b/2)2 ≤ a2+b2/2. Those vectors are supposed to be in column form. Homework Equations |<u,v>| ≤||u|| ||v||, and the fact that inner product here is defined by dot product (so <u,v> = u\cdotv)...

is \frac{x-1}{x}<ln(x)<x-1 valid for 0<x<1 I think it is I just want to get a second opinion.

Showing the inequality holds for an interval (?) Homework Statement Hi, my homework question is: Show that the inequality \sqrt{2+x}<2+\frac{x}{4} holds \forallx\in[-2,0] Homework Equations The Attempt at a Solution I tried using IVT or bisection method, but they are just for...

I've been reading a book called Superfractals, and I'm having trouble with a particular proof: Definitions: The distance from a point x \in X to a set B \in \mathbb{H}(X) (where \mathbb{H}(X) is the space of nonempty compact subsets of X is: D_B(x):=\mbox{min}\lbrace d(x,b):b \in B\rbrace The...

Hi. I just saw on wikipedia that natural logarithm has such a property: [x/(1+x)] < ln (1 + x) < x (http://en.wikipedia.org/wiki/Natural_logarithm) Can anyone pls tell me how to prove this? Proving [x/(1+x)] and ln (1 + x) less than 'x' is easy.. But how abt [x/(1+x)] < ln (1 + x)...

Homework Statement Please see below... Homework Equations Please see below... The Attempt at a Solution Hi. This question is on geometry with circle and triangle. I am stuck only on 2 parts of the solution and not the whole solution... Thank you...

Is there a way to do this without differentiation? \left(a+b\right)^{p} \leq a^{p}+b^{p} 0<p<1 and a,b\geq 0 pulling the a out of the the first part and dividing by it to get \left(1+\frac{b}{a}\right)^{p}\leq 1+\frac{b}{a}^{p} This seems like the way to go but am stuck. Any...

In an example in my textbook, it says the following: "If -1 ≤ x ≤ 1, then 0 ≤ x2 ≤ 1. " Can someone explain to me how to move from the first statement to the second statement please? I'm not quite sure how the -1 turned into a 0...

Homework Statement Prove that \sum_{k=0}^n {3k\choose k}\ge \frac{5^n-1}{4}Homework Equations {3k\choose k}= \frac{(3k)!}{k!(2k)!}The Attempt at a Solution I tried using the induction principle, but... Here my attempt: For n=0 1>0 ok Suppose that is true for n, i.e.: \sum_{k=0}^n...

Hello Friends, I at a loss to understand the parts of the following proof: For any positive ineteger n, prove that: (1+1/n)^n < (1+1/n+1)^n+1 a, b positive real numbers such that a < b Proof: b^n+1 - a^n+1 = (b-a)(b^n+ab^n-1+...+a^n) I could not understand the following part: By...

This may seem trivial, but for some reason I am having trouble with it. For a and b in the complex plane, I am trying to prove the following: |a|^2+|b|^2 >= |(a+b)/2|^2 I need this for part of a larger proof.

Homework Statement I already have the solutions, but I am not sure what the solutions are trying to say. http://img194.imageshack.us/img194/2595/unledlvc.jpg So in I don't understand this, we have n > \frac{1}{\epsilon} and If (and I am guessing we really want this to...

Homework Statement Determine the set of positive values of x that satisfy the following inequality: (1/x) - (1/(x-1)) > (1/(x-2)) a) (0, 1) union (2^1/2, 2) b) (0, 1/2) union (1, 2) c) (1/2, 1) union (2^1/2, 2(2^1/2)) d) (0, 2^1/2) union (3/2, 2) e) (1, 2^1/2) union (2, 2(2^1/2))...

In this link is a part of a book on approximations of functions. http://books.google.com/books/about/An_Introduction_to_the_Approximation_of.html?id=VTW2cmjC43YC I'd be thankful if someone would explain how the inequality near the top of page 17 was gotten.

Homework Statement Why is \langle p^2\rangle >0 where p=-i\hbar{d\over dx}, (noting the ***strict*** inequality) for all normalized wavefunctions? I would have argued that because we can't have \psi=constant, but then I thought that we can normalize such a wavefunction by using periodic...

Homework Statement Determine the values of z \in \mathbb{C} for which |z+2| > 1 + |z-2| holds. Homework Equations Nothing complicated I can think of. The Attempt at a Solution For real values this holds for anything greater than 1/2. If I could figure out the boundaries of the...

a,b,c,d\in\mathbb{R^{+}}\;\;,a+b+c+d=1. Then prove that \left( a+\dfrac{1}{b}\right).\left(b+\dfrac{1}{c}\right).\left(c+\dfrac{1}{a}\right)\geq \left(\dfrac{10}{3}\right)^3 Anyone an idea on how to start with this exercise?

Young's Inequality can be restated as: s^(x)t^(1-x)<=xs + (1-x)t where s,t>=0 and 0<x<1. Basically I've been asked to prove this. I've been fiddling about with it for a couple of hours to no avail. I've tried to substitute t=e^u and s=e^v and then use partial differentiation w.r.t to...

Homework Statement What I want to show is this: ∫|x+y| ≤ ∫|x| + ∫|y| Homework Equations |x+y| ≤ |x| + |y| The Attempt at a Solution So I thought if I used the triangle inequality I could get to something along the lines of: Lets g belong to the real numbers ∫|x+y| =...

Hello all, the problem I have is the following: Suppose f \in C^1(0,1) and f(0) = 0, then f^2(x) \le \int_0^1 f^2(x) dx, but I was wondering if 1 is the best constant for the inequality. In other words, how do I determine the best bound for f^2(x) \le K \int_0^1 f^2(x) dx...

Homework Statement Suppose that w is a complex number which is not both real and \left\lfloorw\right\rfloor\geq1 (the absolute value of w). Verify that Re[(1-w^{2})^{1/2}+iw]>0. Homework Equations The Attempt at a Solution I attempted to solve this problem by dividing it into...

Homework Statement 2x-1 _____ > 0 5x+3 Homework Equations The Attempt at a Solution Just wondering, my teacher taught us that youre only supposed to look at what makes the denominator = 0, and don't look at the numerator because it has no affect on anything. So, if i...

Can Someone please solve this inequality! Homework Statement ((2x)3^x)/(x+1) < 0 Thanks is advance! Homework Equations The Attempt at a Solution

Homework Statement This is part of a question on absolute convergence on series. The following equation is given as a hint. It says that before answering the question on series I should prove that |xy| <= 1/2(|x|^2 + |y|^2) for any x,y ε R Homework Equations The Attempt at a...

Homework Statement > a[1], a[2], a[3], .. , a[n] are arbitrary real numbers, prove that; abs(sum(a[i], i = 1 .. n)) <= sum(abs(a[i]), i = 1 .. n) Homework Equations The Attempt at a Solution I have uploaded my attempt as a pdf file, since I'm not too familiar with the...

p<q, r<s, and r<q. Which of the following statements must be true? I. p<s II. s<q III. r<p The correct answer could be either one statement, a combination of statements, or none of the statements. Came across this question while helping some high school students prepare for their SATs...

Homework Statement 4x5-16x4+9x3+23x2-15x-9 > 0 Homework Equations Synthetic division PQ Rule? The Attempt at a Solution Don't know how or where to begin

Hello I am doing a calculus proof with epsilon-delta and I am trying to say the following: -1\leqsin x\leq1 and now I want to get (sin x )^2 ...so can you just square all sides of the inequality like this: (-1)^2\leq(sin x)^2\leq(1)^2 ?? According to the rule for inequalities...

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...

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...