Inequality Definition and 1000 Threads

Homework Statement Homework EquationsThe Attempt at a Solution i just straight up applied am gm ##\frac {x}{y+z} + \frac{y}{x+z} + \frac{z}{x+y} \geq 3(\sqrt[3]{\frac{1}{(y+z)(x+z)(x+y)}}) ## so the denominator is which i had to maximise ## x^2(y+z) + y^2(x+z) +z^2(x + y) + 2\\...

1)Prove without using AM-GM :$$\frac{ab}{c}+\frac{ac}{b}+\frac{bc}{a}\geq a+b+c$$...... a,b,c >02) Prove without using contradiction : $$a\leq b\wedge b\leq a\Longrightarrow a=b$$

given a>0 find b>0 such that: $$\sqrt{(x-1)^2+(y-2)^2}<b\Longrightarrow |xy^2-4|<a$$

I'm pretty sure that the following is true, but I don't see an immediate compelling proof, so I'm going to throw it out as a challenge: Let A,A', B, B' be four real numbers, each in the range [0,1]. Show that: AB + AB' + A'B \leq A' B' + A + B (or show a counter-example, if it's not true)...

Hello. I am trying to prove that the uncertainty in energy for a normalized state limits the speed at which the state can become orthogonal to itself. The problem is number 2 on https://ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/assignments/MIT8_05F13_ps6.pdf Having issues...

Using high school mathematics prove the following inequality: $$\sqrt{a_{1}^2+...+a_{n}^2}\leq\sqrt{(a_{1}-b_{1})^2+...+(a_{n}-b_{n})^2}+\sqrt{b_{1}^2+...+b_{n}^2}$$

Hello all, Jensen's inequality says that for some random x, f(E[x])≤E[f(x)] if f(x) is convex. Is there any generality that might help specify under what circumstances this inequality is...equal? Thanks

Homework Statement F(x,y,z)=4x i - 2y^2 j +z^2 k S is the cylinder x^2+y^2<=4, The plane 0<=z<=6-x-y Find the flux of F Homework Equations The Attempt at a Solution What is the difference after if I change the equation to inequality? For example : x^2+y^2<=4, z=0 x^2+y^2<=4 , z=6-x-y...

I began reading Mehran Kardar's Statistical Physics of Particles and about halfway through the first chapter, there was a discussion on the second law of thermodynamics. He makes no mention of the old tenet that 'the total entropy in the universe must always increase' (I'll refer to this as the...

Prove: $$\sqrt{a^2+b^2+c^2}\leq\sqrt{(x-a)^2+(y-b)^2+(z-c)^2}+\sqrt{x^2+y^2+z^2}$$

Homework Statement Hello! The task is to express the exact answer in interval notation, restricting your attention to -2π ≤ x ≤ 2π. Homework Equations The given inequality: cos(x) ≤ 5/3 The Attempt at a Solution I have only one doubt here, and I don't see my mistake. I see that if cos(x)...

How do I find the range of [(4 - 4x^2)/(x^2 + 1)^2] > 0 algebraically? Do I set the numerator to 0 and solve for x? Do I set the denominator to 0 and solve for x?

Hi. I wonder if following thought experiment (which is most probably impossible to be put into practice) could have any implications concerning interpretations of QM. Consider five parties A, B, C, D and E, lined up in that order and with no relevant relative motion. No pair of them have ever...

Hi, I am reading a paper and in this equation is given. I don't quite follow how they end up with the last (Uw-UL)<a-B. If I do it myself I get the inequality sign wrong. Any help? Thx

Prove, that for any triangle:\[ \sum_{cyc}\sin A - \prod_{cyc}\sin A \ge \sum_{cyc}\sin^3 A \]

Homework Statement Consider a real valued function f which satisfies the equation f (x+y) = f (x) . f (y) for all real numbers x and y. Prove: f ((x + y) / 2) ≤ 1/2 (f(x) + f(y)) Homework Equations Not sure The Attempt at a Solution Please give me a hint to start solving this question. I...

Homework Statement For a,b,c,d >1, Show that (a+1)(b+1)(c+1)(d+1) < 8(abcd+1) Homework Equations How to show this? The Attempt at a Solution I could show for two variables, (a+1)(b+1)<2(ab+1). Tried C-S, AM-GM inequalities in different form and variable transformations. But still no result...

Let $0 \le a,b,c \le 1.$ Prove the inequality:$\sqrt{a(1-b)(1-c)}+ \sqrt{b(1-a)(1-c)}+\sqrt{c(1-a)(1-b)} \le 1 + \sqrt{abc}$

Acute triangle ABC Prove :Sin A +Sin B +Sin C>Cos A + Cos B + Cos C

Prove, that$\sqrt{1+\sqrt{2+\sqrt{3+...+\sqrt{2006}}}}<2.$

Everything I've seen about Bell's inequality has had the setup of 120 degree angles between the axis of measurements. The experiment then proves that the basic hidden variable theory can't be true. But the actual measurement has always been told to me as a 0.5 correlation. 50% of the time the...

Homework Statement "Given: ##a,b,c∈ℤ##, Prove: If ##2a+3b≥12m+1##, then ##a≥3m+1## or ##b≥2m+1##." Homework Equations ##P:a≥3m+1## ##Q:b≥2m+1## ##R:2a+3b≥12m+1## The Attempt at a Solution Goal: ##~(P∨Q)≅(~P)∧(~Q)⇒~R## Assume that ##a<3m+1## and ##b<2m+1##. Then...

Hi all, https://en.m.wikipedia.org/wiki/Bonse's_inequality It seems to me that the inequality can be true for higher powers (if not any given higher power), for an appropriately higher (lower) bound for "n". Any thoughts, proofs, counter proofs your insights are appreciated. In particular, I...

I believe this is probably a high level undergraduate question, but i could easily be underestimating it and it's actually quite a bit higher than that. I'm reading the Prime number theorem wikipedia page and I'm in part 4 under Proof sketch where sometime down they give in inequality: x is a...

Homework Statement Prove that for any naturam number n > 1 : \frac{1}{n+1} + \frac{1}{n+2} + \frac{1}{n+3} + ... + \frac{1}{2n} > \frac{13}{24} Homework Equations Not sure The Attempt at a Solution \frac{1}{n+1} + \frac{1}{n+2} + \frac{1}{n+3} + ... + \frac{1}{2n} > \frac{1}{2n} +...

Bell inequality in page 171 of https://www.scientificamerican.com/media/pdf/197911_0158.pdf is ##n[A^+B^+] \le n[A^+C^+]+n[B^+C^+]## In page 174 we can see that this causes linear dependency according to angle. How to derive this? Let us suppose that angle between ##A^+## and ##B^+## is 30°...

Homework Statement If a,b,c,d,e>1 then prove that a^2/(c-1)+b^2/(d-1)+c^2/(e-1)+d^2/(a-1)+e^2/(b-1)=>20 The Attempt at a Solution Given a,b,c,d,e are roots of a polynomial equation of a degree 5 then x^2/(x-1)+x^2/(x-1)+x^2/(x-1)+x^2/(x-1)+x^2/(x-1)=>20 5 x^2/(x-1)=>20 x^2/(x-1)=>4 x^2=>4x-4...

Homework Statement Homework EquationsThe Attempt at a Solution Hi How do I go about showing ##0 \leq \frac{2x}{\pi} \leq sin x ##? for ## 0 \leq x \leq \pi /2 ## I am completely stuck where to start. Many thanks. (I see it is a step in the proof of Jordan's lemma, but I'm not interested in...

Homework Statement please see attached, I am stuck on the second inequality. Homework Equations attached The Attempt at a Solution I have no idea where the ##2/\pi## has come from, I'm guessing it is a bound on ##sin \theta ## for ##\theta## between ##\pi/4## and ##0## ? I know ##sin...

Prove the inequality: \[\left | \cos x \right |+ \left | \cos 2x \right |+\left | \cos 2^2x \right |+...+ \left | \cos 2^nx \right |\geq \frac{n}{2\sqrt{2}}\] - for any real x and any natural number, n.

Assume f and g are two continuous functions in (a, b). If at the start of the segment I've shown f>g by taking the lim where x ---> a+ and the f ' > g ' for every x in (a,b ) can i say that f >g for all x in (a,b )? is there a theorem for that? that looks intuitively right.

The problem Show that ## 1-x+ \int^x_1 \frac{\sin t}{t} \ dt < 0## for ## x > 1 ## The attempt I rewrite the integral as ##\int^x_1 \frac{\sin t}{t} \ dt < x-1 ## This is about where I get. Can someone give any suggestions on how to continue from here?

Hey! :o Let $\mathbb{K}$ be a fiels and $A\in \mathbb{K}^{p\times q}$ and $B\in \mathbb{K}^{q\times r}$. I want to show that $\text{Rank}(AB)\leq \text{Rank}(A)$ and $\text{Rank}(AB)\leq \text{Rank}(B)$. We have that every column of $AB$ is a linear combination of the columns of $A$, or not...

Homework Statement Hi guys, I would just like someone to go over my method for this derivation/proof ( not sure of the right word to use here). Anyway I think this is right method, but just feel like I am missing something. Could someone please check my method. Thanks in advance. Homework...

Homework Statement Given : (y+2)(y-3) <= 0Homework EquationsThe Attempt at a Solution Now, I have y-3 <= 0 or y+2 <= 0 Hence, y <= 3 or y <= -2 But how is correct? I think is wrong because y <= -2. Can someone please clarify?

Homework Statement Prove the following facts about inequalities. [In each problem you will have to consider several cases separately, e.g. ##a > 0## and ##a = 0##.] (a) If ##a \leq b##, then ##a + c \leq b + c##. (b) If ##a \geq b##, then ##a + c \geq b + c##. (c) If ##a \leq b## and ##c \geq...

Proof: If either x or y is zero, then the inequality |x · y| ≤ | x | | y | is trivially correct because both sides are zero. If neither x nor y is zero, then by x · y = | x | | y | cos θ, |x · y|=| x | | y | cos θ | ≤ | x | | y | since -1 ≤ cos θ ≤ 1 How valid is this a proof of the...

Homework Statement Let ##a,b,c## be positive integers and consider all the quadratic equations of the form ##ax^2-bx+c=0## which have two distinct real roots in ##(0,1)##. Find the least positive integers ##a## and ##b## for which such a quadratic equation exist. Homework EquationsThe Attempt...

Prove the inequality: $\sqrt{a(1-b)(1-c)}+\sqrt{b(1-a)(1-c)}+\sqrt{c(1-a)(1-b)} \le 1 + \sqrt{abc}, \;\;\;\;a,b,c \in [0;1].$

$$ \sin{(\pi x)}>\cos{(\pi \sqrt{x})} $$ I don't know how to solve this. I would really appreciate some help. I tried to do something, but didn't get anything. If I move cos to the left side, I can't apply formulas for sum. Since arguments of sin and cos have $$ \pi $$, I think there is no way...

$given :\,\,$ $a,b,c\geq1$ $prove:$ $(1+a)(1+b)(1+c)\geq 2(1+a+b+c)$

Homework Statement show that 1-t^2/2 <=cos(t) <=1 for 0<=t<=1 Homework Equations Trigonometry knowledge The Attempt at a Solution I don't know how to relate t with cos(t), and I also try to find out cos(1), but there is no result, so how can I start with this problem.

So I am trying to solve a simple rational inequality: ##\sqrt{x} < 2x##. Now, why can't I just square the inequality and go on my way solving what results? What precisely is the reason that I need to be careful when squaring the square root?

Hello, if we consider the vector spaces of integrable real functions on [a,b] with the inner product defined as: \left \langle f,g \right \rangle=\int _a^bf(x)g(x)dx the Cauchy-Schwarz inequality can be written as: \left | \int_{a}^{b} f(x)g(x)dx\right | \leq \sqrt{\int_{a}^{b}f(x)^ 2dx}...

If the system of inequalities y ≥ 2x + 1 and y> x/2-1 is graphed in the xy-plane above, which quadrant contains no solutions to the system? A) Quadrant II B) Quadrant III C) Quadrant IV D) There are solutions in all four quadrants. I thing the answer is D . But book says that it is C. I...

The polynomial: $P(x) = 1 + a_1x +a_2x^2+...+a_{n-1}x^{n-1}+x^n$ with non-negative integer coefficients has $n$ real roots. Prove, that $P(2) \ge 3^{n}$

Homework Statement $$ \left | \frac{z}{\left | z \right |} + \frac{w}{\left | w \right |} \right |\left ( \left | z \right | +\left | w \right | \right )\leq 2\left | z+w \right | $$ Where z and w are complex numbers not equal to zero. 2.$$\frac{z}{\left | z \right | ^{2}}=\frac{1}{\bar{z}}$$...

Show that if $a,\,b$ and $c$ are the lengths of the sides of a right triangle with hypotenuse $c$, then $$\frac{(c − a)(c − b)}{(c + a)(c + b)}\le 17 − 12\sqrt{2}$$

Mass of a book of type X = 40g; Mass of a book of type Y=80g; The total mass of n books of type X and one book of type Y is less than 200g; i. Write down an inequality containing the variable n only. ii. Solve the above inequality for n and write down the maximum possible value for n So how...

just starting out with proofs... i tried manipulating the right side of the inequality, but i don't see why it's equal to (n+1)!