Inequality Definition and 1000 Threads

Homework Statement then what are the operations that maintain the Inequality and what are the operations that don't? Homework Equations The Attempt at a Solution clearly addition and subtraction maintains it ,and so does multiplication and division by any number other than...

I'm trying to figure out how to prove this inequality: I know it's true (by graphing), but what's an algebraic way to prove it? 1+\frac{1}{3x^2}< x \tan \frac{1}{x} < \frac{1}{\sqrt{1-\frac{2}{3x^2}}} Thanks

Homework Statement I am trying to prove that \frac{1}{512} \left[101-(1-\alpha ) \gamma ^n \left(2 (37+64 \alpha )+27 (1-\alpha ) \gamma ^n\right) (1-\delta )-\frac{128 n (1-\alpha ) \alpha (1-\gamma ) \gamma ^{-1+n} (1-\delta )}{\alpha +\delta -\alpha \delta }\right]\geq0 for...

Homework Statement Prove without computation that 2<Integral[0,2] (1+x^3)<6 The Attempt at a Solution I know there is a theorem which says that if a function is bounded by two constants, then the integral of the function is also bounded by the integrals of the two functions. However...

Homework Statement Graph the following inequality in the complex plane: [FONT="Courier New"]|1 - z| < 1 2. The attempt at a solution In order to graph the inequality I need to get the left side in the form [FONT="Courier New"]|z - ...| [FONT="Courier New"]|1 - z| < 1 |(-1)z + 1| < 1 |-1(z...

Homework Statement Solve the Inequality: (3x-7)/(x+2)<1 Homework Equations The Attempt at a Solution Cross Multiply: x+2>3x-7 Simplify: 9>2x Simplify More: 9/2>x My Answer: (-∞, 9/2) I put this as my answer but the answer is really (-2, 9/2) Can someone explain to me why this is? I know you...

Homework Statement Let V be a vector space with inner product <x,y> and norm ||x|| = <x,x>^1/2. Prove the Cauchy-Schwarz inequality <x,y> <= ||x|| ||y||. Hint given in book: If x,y != 0, set c = 1/||x|| and d = 1/||y|| and use the fact that ||cx ± dy|| >= 0. Here...

Is inequality bad for society as a whole?

Homework Statement Prove that ||a|-|b||\leq |a-b| for all a,b in the reals Homework Equations I know we have to use the triangle inequality, which states: |a+b|\leq |a|+|b|. Also, we proved in another problem that |b|\leq a iff -a\leqb\leqa The Attempt at a Solution Using the...

Question: I need to prove this inequality: Where x,y,x are non-negative and x+z<=2: (x-2y+z)^2 >= 4xz -8y. My attempt: I thought maybe choosing x as 0 and z as 0 will and then solving for y... but that only yields y+2 >= 0, which isn't really a solution, since I can't choose numbers...

Homework Statement Prove that if a,b > 1, then a+b < 1+ab The Attempt at a Solution Just want to know if this makes sense: first let a+b < 1+ab become 1<(1+ab)/(a+b) ==> 0<(1+ab-(a+b))/(a+b). Factoring the numerator: 0<(1-a+ab-b)/(a+b) ==> 0<(1-b)+a(b-1)/(a+b) So the next...

Hi, I was reading Heinz Pagels' description of the nail gun experiment in the chapter about "Bell's Inequality" from his book, The Cosmic Code: Quantum Physics as the Langauge of Nature, 1982, pp. 160-176. He describes the record of hits and misses after "turning polarizer A clockwise by...

Got some exciting news from a PF Mentor: And the actual paper: http://iopscience.iop.org/1367-2630/12/12/123007 Violation of Leggett inequalities in orbital angular momentum subspaces J Romero, J Leach, B Jack, S M Barnett, M J Padgett and S Franke-Arnold J Romero et al 2010 New J...

Homework Statement Let f(z) be an entire function such that |f(z)| less that or equal to R whenever R>0 and |z|=R. (a)Show that f''(0)=0=f'''(0)=f''''(0)=... (b)Show that f(0)=0. (c) Give two examples of such a function f. Homework Equations The Attempt at a Solution...

Homework Statement let n\inN To prove the following inequality na^{n-1}(b-a) < b^{n} - a^{n} < nb^{n-1}(b-a) 0<a<b Homework Equations The Attempt at a Solution Knowing that b^n - a^n = (b-a)(b^(n-1) + ab^(n-2) + ... + ba^(n-2) + a^(n-1) we can divide out (b-a) because b-a #...

Hello, i've met during problem solving with inequality \max\{A+B,C\}\le\max\{A,C\}+\max\{B,C\} where A,B and C are real numbers. I don't know whether it holds, but I need to prove that. Thanks for reply...

Homework Statement find n0,c1,c2 for which the following is true: c1 nb <=(n-a)b<=c2(n-a)b , for all n > n0Homework Equations http://en.wikipedia.org/wiki/Binomial_theorem" ?The Attempt at a Solution c1 nb <=(n-a)b<=c2(n-a)b c1 nb <=nb-nb-1a+nb-2a2-...-ab<=c2nb c1<=1-a/n + a2/n2-...

Homework Statement For f nonnegative and continuous on [0,1], prove. \left( \int f \right) ^2 < \int f^2 With the limits from 0 to 1. Homework Equations The Attempt at a Solution I was trying to use Upper sums, i.e. \inf \sum \Delta x_i M_i(f^2) = \inf \sum \Delta x_i...

Prove that in any triangle ABC with a sharp angle at the peak C apply inequality:(a^2+b^2)cos(α-β)<=2ab Determine when equality occurs. I tried to solve this problem... I proved that (a^2+b^2+c^2)^2/3 >= (4S(ABC))^2, S(ABC) - area but I don't know prove that (a^2+b^2)cos(α-β)<=2ab :(...

Is it true in general that: |\int f(x)dx| < \int |f(x)|dx Not sure if "Triangle Inequality" is the right word for that, but that seems to be what's involved.

Homework Statement Prove that 2 \leq 1+ \sum(m=1 to n) 1/m! \leq 1 + \sum (m=1 to n) (1/(2^(m-1))) < 3 The Attempt at a Solution I've proved by induction that 2m-1 \leq m!, so it just follows that 1 + (1/(2 ^ (m-1))) \geq 1 + (1/m!), and their sums are the same inequality...

suppose that g:[0,1] \rightarrow \re is continuous, g(0)=g(1)=0 and for every c \in (0,1), there is a k > 0 such that 0 < c-k < c < c+k < 1 and g(c)=\frac(1}{2} (g(c+k)+g(c-k)). Prove that g(x) = 0 for all x \in [0,1] Hint: Consider sup{x \in [0,1] | f(x)=M } where M is maximum of f on [0,1]...

1. Suppose that f: (a,b) --> R is convex. Prove Jensen's inequality: if x1,...,xn\in(a,b) and c1,...,cn >= 0 s.t. \sum(c_j)f(x_j) >= f(\sum((c_j)(x_j)) both summations from j = 1 to n 2: Convex: whenever x1, x2 \in(a,b) and 0 <= c <= 1, we have cf(x1) + (1 + c)f(x2) >= f(cx1 + (1-c)x2)...

This was already posted by someone else but an answer wasn't received so I thought I'd repost. Any help is appreciated. Homework Statement Use the Maclaurin series for cosx and the Alternating Series Estimation Theorem to show that \frac{1}{2} - \frac{x^2}{24} < \frac{1-cosx}{x^2} <...

Homework Statement Use the Maclaurin series for cosx and the Alternating Series Estimation Theorem to show that \frac{1}{2} - \frac{x^2}{24} < \frac{1-cosx}{x^2} < \frac{1}{2} Homework Equations cosx = 1 - \frac{x^2}{2} + \frac{x^4}{4} - \cdot \cdot \cdot = \sum_{n=0}^\infty...

Homework Statement Prove that if n is a natural number and is greater/equal to 4, then 2n<n!, and show that 2n is less/equal to 2((n-1)!) follows. The Attempt at a Solution I'm thinking I just need to use induction on n for the first part, where I get the inequality (n+1)! = n!*(n+1) >...

Homework Statement d(x,y)=(a|x_1-y_1|^2+b|x_1-y_1||x_2-y_2|+c|x_2-y_2|^2)^{1/2} where a>0, b>0, c>0 and 4ac-b^2<0 Show whether d(x,y) exhibits Triangle inequality? Homework Equations (M4) d(x,y) \leq d(x,z)+d(z,y) (for all x,y and z in X) The Attempt at a Solution I...

Homework Statement Show that 1/2(1-cos1)\leq\int\intsinx/(1+(xy)4)dxdy\leq1 on the area 0\leqx\leq1, 0\leqy\leq1. Homework Equations Mean Value Inequality: m*A(D)\leq\int\intf(x,y)dA\leqM*A(D), where m is the minimum and M is the maximum on the interval. The Attempt at a Solution...

Homework Statement Use your knowledge of exponents to solve \frac{1}{2^x} > \frac{1}{x^2} Homework Equations The Attempt at a Solution x^2 > 2^x Then I am stuck. I know they intersect at x = 2.

Homework Statement 1/x <= 4 Homework Equations The Attempt at a Solution I initially converted 1/x back to x^-1 which gave me the answer x <= 1/4 which makes sense, but I should also get x < 0 which I'm not sure about how to get via solving? Also is converting 1/x to x^-1 the...

Suppose f, g:[a,b]->R are bounded & g(x)<=f(x) for all x in [a,b] for P a partition of [a,b], show that L(g,P)<=L(f,P) I don't know whether I should show by cases since I don't know the monotonicity of the both functions f and g. It seems like that the graphs of both functions have to behave...

Hi everyone Today during problem session we had this seemingly simple exercise, but I just can't crack it: We should give an example of an x \in \ell^2 with strict inequality in the Bessel inequality (that is an x for which \sum_{k=1}^\infty |<x,x_k>|^2 < ||x||^2, where (x_k) is an orthonormal...

Homework Statement prove that for all x>0 Homework Equations -1 \leq sin t \leq 1 The Attempt at a Solution the area under the graph is increasing as x increases also, i tried to write it the sigma way: then take the limit as n-->infinity i got stuck trying to figure out how to...

Homework Statement For what values of the variable x does the following inequality hold: \frac{4x^2}{(1-\sqrt{1+2x})^2}<2x+9 Homework Equations The Attempt at a Solution Maybe some hints for me to begin.

I was wonder what conditions a and b have to be for each inequality in order to satifsy the equality?

Homework Statement If x is real and y=\frac{x^2+4x-17}{2(x-3)}, show that |y-5| \geq 2Homework Equations The Attempt at a Solution Sorry... Absolutely no idea. I tried to substitute y into the left side to prove that -2 \leq y - 5 \leq 2 but I can't. Anything I should know to do this?

Homework Statement P(AUB) <= P(A) + P(B) Homework Equations The Attempt at a Solution I can't understand the intuition behind this property. It's not a homework assignment, it was just something that came up in class. Thanks, M

Homework Statement If a, b and c are distinct positive numbers, show that 2 (a^3 + b^3 + c^3) > a^2b + a^2c + b^2c + b^2a + c^2a + c^2b Homework Equations The Attempt at a Solution I have tried to expand from (a+b+c)^3 > 0, also tried (a+b)^3 + (b+c)^3 + (c+a)^3 > 0, and then...

Suppose I have three vectors a,b and c in R^d , And, I have that a.b < c.b(assume Euclidean inner product). What are the ways to visualize relation between a,b and c geometrically? I realize this is slightly open-ended, but am looking for insight here. Thanks in advance. PS: I have a thought...

I expanded (x+y),(x+y) and got x^2+y^2 > 2xy then replaced 2xy with 2|x,y| but now I'm stuck. I need to get it to ||x+y|| <= ||x|| + ||y||. Am I close?

Homework Statement A rectangle solid is to be constructed with a special kind of wire along all the edges. The length of the base is to be twice the width of the base. The height of the rectangular solid is such that the total amount of wire used (for the whole figure) is 40cm. Find the...

Homework Statement Let r_{1}, r_{2}, ... , r_{n} be strictly positive numbers. Show that the inequality (1+R_{G})^{n} \leq V is true. Where R_{G} = (r_{1}r_{2}...r_{n})^{1/n} and V= \Pi_{k=1}^{n} (1+r_{k}) Homework Equations The Attempt at a Solution I've...

Homework Statement Show that \forall a,b \in R: \left|ab\right|\leq\frac{1}{2}(a^{2}+b^{2}) Homework Equations Triangle Inequality seems to be useless. The Attempt at a Solution (a+b)^{2}=a^{2}+b^{2}+2ab 2ab=(a+b)^{2}-(a^{2}+b^{2})...

Homework Statement let r_{1}, r_{2}, ... , r_{n} be strictly positive numbers. Suppose an investment of one dollar at the beginning of the year k grows to 1+r_{k} at the end of year k (so that r_{k} is the "return on investment" in year k). Then the value of an investment of one dollar at...

Pretty much knows the triangle inequality. \left| a + b \right| \le \left| a \right| + \left| b \right| I was reading a source which asserted the following extension of the triangle inequality: \left| a + b \right|^p \le 2^p \left(\left| a \right|^p + \left| b \right|^p\right) This is...

[FONT="Arial Black"]hi every body show if Xn→x then lXnl→lxl hint use trangle inequality 2/ show if lXnl→0 then Xn →0 show by example that lXnl fore all n in N MAY CONVERGE and Xn may not converge

Homework Statement prove that llal-lbll\leqla-bl Homework Equations Triangle inequality lx+yl\leqlxl+lyl The Attempt at a Solution Let a=(a-b)+b By using the triangle inequality we get lal-lbl\leqla-bl Then from here I am not sure what I can do. I would like to say on the left...

Homework Statement Prove a)5 < 51/2 + 51/3 + 51/4 b) n > n1/2 + n1/3 + n1/4 for all ints n>8 Homework Equations The Attempt at a Solution i tried the AM GM inequality and found 51/2 + 51/3 + 51/4 > 3(513/36) what further can i do? can anyone please help me out??

Homework Statement The question says to find a proof for Cauchy's Inequality and then the Triangle Inequality. This is an elementary linear algebra class I'm doing, so I can't use inner products or anything. Homework Equations The Attempt at a Solution I got the proofs using algebra, but I'm...

Homework Statement Show \sum_{i=1}^n \frac{1}{i^2} \leq 2 - \frac{1}{n} with induction on n. I'm pretty rusty on induction (not that I was very good at it to being with), so I mostly wanted to know if I'm on the right track, and if this is a way towards a valid proof. Homework Equations The...