Inequality Definition and 1000 Threads

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...

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 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...

Homework Statement Prove that for all numbers a, b, c, d: if 0 \leq a < b and 0 \leq c < d then ac < bd. This is problem 5 from chapter 1 of Michael Spivak's "Calculus", 4th Edition. It is the text for my real analysis course. I should also mention that this is not a homework problem...

Hi, Does the following inequality hold regarding the product of 2 matrices A and B: p(AB) <= p(A)p(B), where p denotes the spectral radius. Thanks!

While this is not technically an assignment for any particular class (that I'm aware of, at least), I think the nature of this problem makes it suitable for this forum. Please, inform me if I should direct my question elsewhere. Find x>3 such that ln(x)<x^0.1 (hint: The number is "huge")...

Hi, I am working on a calculus of variations problem and have a general question. Specifically, I was wondering about what kind of constraint functions are possible. I have a constraint of the form: f(x)x - \int_{x_0}^x f(z) dz \leq K If I had a constraint that just depends on x or...

Let be k \leq n poitive integers. How to show that \left (1+\frac1 n \right)^k \leq 1 + \frac{ke}{n} . It seems to me that it has something to do with Bernoulli's inequality. Thank you in advance!

Homework Statement I am attempting to show that -x \leq sin(x) \leq x for x>0 and thus \int^1_0 nxsin(\frac{1}{nx})dx converges to 1. Homework Equations I know that I need to use the fact that I have shown that the limit as T tends to infinity of \int^T_1 \frac{cos(x)}{\sqrt{x}}dx...

Homework Statement Homework Equations The Attempt at a Solution

I want to find value for m for which: 4m2 - 12m > 0 Say I do this algebraically: 4m(m-3) > 0 so m > 0 or m > 3 The answer however is 0 < m and m > 3, I know this as a fact as I have looked graphically. So, my question is, when done algebraically, how do I get 0 < m instead of m...

Homework Statement Show from definition that if f is measurable on [a,b], with m<=f(x)<=M for all x then its lebesgue integral, I, satisfies m(b-a)<=I<=M(b-a) Homework Equations The Attempt at a Solution I know that the definition is that f:[a,b]->R is measurable if for each t...

Homework Statement Prove the Triangle Inequality Theorum using the coordinate system. Homework Equations The corners of the triangles will be at (x1,y1), (x2, y2), (x3,y3) The Attempt at a Solution The proof that I know is proving that |x+y|<=|x|+|y|: -|x|<x<|x|, and...

Hi I'm in the process of trying to understand the proof to bessel's equality and inequality and I am stuck, I have got to the line http://img141.imageshack.us/img141/396/besselsequality.jpg Uploaded with ImageShack.us and I'm not entirely sure how it equates to the next line but according to...

My mathematical methods for theoretical physics course recently began looking at linear vector spaces. We defined the Banach and Hilbert Spaces and proved the Cauchy-Shwarz Inequality. There's one step in this proof that I can't really follow (in red): consider: w=x+uy (i'll drop the...

Homework Statement Hi Guys, I try to find the range for parameters phi1 and phi2 were the autoregressive process below is stationary. We have the process X(t)+phi1*X(t-1)+phi2X(t-2)=Epsilon(t) (1) Homework Equations We get the characteristic polynomial F(z)=z^2+phi1*z+phi2 (2) The...

I have the following question on metric spaces Let (X,d) be a metric space and x1,x2,...,xn ∈ X. Show that d(x1, xn) ≤ d(x1, x2) + · · · + d(xn−1, xn2 ), and d(x1, x3) ≥ |d(x1, x2) − d(x2, x3)|. So the first part is simply a statement of the triangle inequality. However, the metric...

Homework Statement Prove the following inequality: \frac{1}{6}\leq\int_{R}\frac{1}{y^{2}+x+1}\chi_{B}(x,y)dxdy\leq\frac{1}{2} where B={(x,y)|0\leq (x)\leq (y)\leq1} and R=[0,1]x[0,1] EDIT: The B region should be 0 less than or equal to x less than or equal to y less than or equal to 1...

Homework Statement Show that a2+b2 =>2ab, and hence, if x+y+z=c, show that x2+y2+z2 => 1/3 c2 Homework Equations The Attempt at a Solution How to prove this when we only have unknowns? The only thing i can think of for the first one is a (a+b)2= a2+b2 +2ab, but how to prove that a2+b2 =>2ab...

Homework Statement Given http://www.mathhelpforum.com/math-help/attachments/f33/20928d1298610998-function-msp281219ebge8he857gc6900005ba9285dff0f5h79.gif , find the values of ''a'' for which the value of the function f(x) <= 25/2. The answer is a<= 1/2. Homework Equations The Attempt at a...

So I multiplied the heat equation by 2u, and put the substitution into the heat equation, and get 2uut-2uuxx=(u2)t=2(uux)x+2(ux)2. I`m not sure where to go from there, I can integrate with respect to t, then I would have a u2 under the integral on the left side, but them I`m not sure where to...

for which f: R \rightarrow R such that \forall x,y\in R does | f(y) - f(x) | \mid \leq (y-x)^2 hold

How are these two equal??(equation, inequality) I study discrete mathematics and we are doing combinations at the moment. There is this example in the book(Discrete Mathematics and Combinatorics p. 30 Ex. 1.43) where it states that the number of integer solutions for: x1+x2+x3+x4+x5+x6<10...

hi, while trying to study complex analysis, i have a few problems. i already know that in complex number system, it's impossible for any order relation to exist. but i was confused to this fact when i saw the proof of triangle inequality. ; Let z,w be complex numbers. Then, triangle...

Dear all, Consider the system given by : http://www.freeimagehosting.net/image.php?53f7eed9ce.jpg where we are trying to solve for s and gamma using Newton's method. It turns out to be a simple implementation. Now, what if we need to impose an inequality constraint on the solution s : one...

Homework Statement If x,y\in R and x+y=1.then find max. and Min. value of (x^3+1)(y^3+1) (Without using calculus) Homework Equations here x+y=1 and (x^3+1)(y^3+1) The Attempt at a Solution I have done using Calculus...