Questions on proving and manipulating inequalities.

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48 views

Show that $\displaystyle 2<(1+x)(1+y)(1+z)<\frac{64}{ 27}$

If $x,y,z>0$ and $x+y+z=1$ then show that $\displaystyle2<(1+x)(1+y)(1+z)<\frac{64}{27}$. I have solved the right hand first using AM-GM inequality, $\displaystyle\frac{1+x+1+y+1+z}{3} ...
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3answers
48 views

Inequalities - x^2 - 1/2 x - 5 < 0 ; why is x > 2 1/2?

Question : $$\text{ find the set of values of }x \text{ for which } $$ $$10 + x - 2x^2 < 0$$ Answer : $$x < -2$$ $$x > 2\frac{1}{2}$$ EDIT - thanks for the responses. To try and ...
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3answers
28 views

Is $f(x)=\sqrt{(1+x^2)}$, $x \in \mathbb{R}$ a contraction mapping?

We take $x, y \in \mathbb{R}$. Then I say : $\mid \sqrt{(1+x^2)} - \sqrt{(1+y^2)} \mid$ $\le$ $\mid 1+\sqrt{x^2} - 1-\sqrt{y^2} \mid$ $\Leftrightarrow$ $\mid \sqrt{(1+x^2)} - \sqrt{(1+y^2)} \mid$ ...
1
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1answer
92 views

Brownian Motion inequality (related to Dvoretzky-Erdoes test)

i have the following question: Let $B(t)$ be a d-dimeansional Brownian motion $d\ge 3$, and $f$ be a monoton increasing function from the positive reals to the positive reals. Let $A_n=(\exists t\in ...
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5answers
149 views

$2^{50} < 3^{32}$ using elementary number theory

How would you prove; without big calculations that involve calculator, program or log table; or calculus that $2^{50} < 3^{32}$ using elementary number theory only? If it helps you: $2^{50} ...
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1answer
44 views

Why is the total expense divided by the number of pizzas made? [closed]

Could you explain to me why $10+2p$ is divided by $p$ in this solution?
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2answers
59 views

Let $x, y \in\mathbb{R}$. Prove that $|x| + |y| \geq |x+y|$

I need to prove the following result: Let $x, y \in\mathbb{R}$. Prove that $|x| + |y| \geq |x+y|$ I know this is the triangle inequality, but I haven't seen one version that helps me solve this ...
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4answers
62 views

How to solve the inequality $x^4<4x^2$? [on hold]

How do you solve the below inequality? $x^4<4x^2$ My answer is (-2, 2)
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0answers
24 views

Need help solving an inequality [duplicate]

How do you graph $x^4<4x^2$? I need to solve the inequality over a set of real numbers. My answer is $(-2, 2)$
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2answers
90 views

An inequality, which is supposed to be simple

Let $x,y,z\in\mathbb{R}$.Let $xy+yz+xz=1$. Prove:$\displaystyle \frac{x}{\sqrt{x^2+1}}+\frac{y}{\sqrt{y^2+1}}+\frac{z}{\sqrt{z^2+1}}\leq \frac{3}{2}$
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0answers
20 views

Inequality involving symmetric polynomials

Let $\bar x = (x_1, x_2, \dots, x_n)$ and $\bar y = (y_1, y_2, \dots, y_n)$ be non-negative vectors in $\mathbb R^n$, and $\bar z = \bar x + \bar y$. For $1 \leq k \leq n$, define the $k$-th symmetric ...
6
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4answers
83 views

Prove that $2 \le \int_0^1 \ \frac{{(1+x)^{1+x}}}{x^x} \ dx \le 3$

I need some starting ideas, hints for proving that $$2 \le \int_0^1 \ \frac{{(1+x)^{1+x}}}{x^x} \ dx \le 3$$ I already checked that with Mathematica that numerically says that $$\int_0^1 \ ...
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4answers
163 views

Prove $1.43 < \displaystyle \int_0^1 e^{x^2} \mathrm{d}x < \frac{e+1}{2}$

Prove $$1.43<\int_0^1 e^{x^2} \mathrm{d}x<\frac{e+1}{2}$$ What I did; As I have no idea how to approach the left inequality I work with $$\int_0^1 e^{x^2} \mathrm{d}x<\frac{e+1}{2} \iff ...
2
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1answer
24 views

Analogous of Markov's inequality for the lower bound

Consider a positive random variable $X$ and call $E[X]$ its expectation. For any positive $a \in \mathbb{R}$, an upper bound for the probability of $P(X>a)$ is provided by the Markov's Inequality, ...
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3answers
47 views

Show that $\, 0 \leq \left \lfloor{\frac{2a}{b}}\right \rfloor - 2 \left \lfloor{\frac{a}{b}}\right \rfloor \leq 1 $

How can I prove that, for $a,b \in \mathbb{Z}$ we have $$ 0 \leq \left \lfloor{\frac{2a}{b}}\right \rfloor - 2 \left \lfloor{\frac{a}{b}}\right \rfloor \leq 1 \, ? $$ Here, $\left \lfloor\,\right ...
0
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2answers
21 views

Algebra. Modeling with one-variable equations and inequalities

I'm solving problems on Khan Academy.I want to know why here 30(1-r) instead of just 30r. Please see the picture below. Thanks
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1answer
21 views

Inequalities and equations - creating sets from quadratic equations.

My question is just making sure that my working is correct and that I understand properly (self teaching, can get confused...) So question : Find the set of values for which $$x^2 -4x-12 < 0$$ ...
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1answer
31 views

What can we say? Variance = Mean

Let X be a random variable with mean and variance equal to $20$. What can you say about $P(0 < X< 40)$? I've tried using chebyshev inequality. We now that $O<X<40$ can be written like ...
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3answers
47 views

Positive values of $x$ that satisfy the inequality $\frac{1}{x}-\frac{1}{x-1}>\frac{1}{x-2}$

Determine the set of positive values of $x$ that satisfy the inequality $$\frac{1}{x}-\frac{1}{x-1}>\frac{1}{x-2}.$$ My attempt: \begin{align} \frac{-1}{x(x-1)} & >\frac{1}{(x-2)} \\[0.1in] ...
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0answers
20 views

Inequalities given specific properties [closed]

Consider a cubic polynomial function $y= f(x)$ with the following properties: $f(x) \geq 0$ only for $x-1$ and $x \geq 3$ when $f(x)$ is divide by $x-4$ the remainder is $50$ Find the equation for ...
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5answers
45 views

If X+Y = 10 (X and Y both are positive) then what is the maximum value of (X^3)*(Y^2)?

I can get to the result by trying different values of X and Y but that is of course time taking. I want to know if there is a better way to get to the result?
2
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2answers
50 views

The smallest $n$ for which the sum of binomial coefficients exceeds $31$

I have a problem with the binomial theorem. What is the result of solving this inequality: $$ \binom{n}{1} + \binom{n}{2} + \binom{n}{3} + \cdots +\binom{n}{n} > 31 $$
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2answers
135 views

Flipping a fraction within an Inequality?

Is it possible to flip a faction within an inequality? Such that: $$\frac13 < x < \frac23$$ becomes, $$3 > \frac1x > \frac32$$
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6answers
94 views

Proving AM-GM for the special case $n=3$

I know the AM-GM inequality and its proof which is relatively complex, though the case for $n=2$ is quite simple. However, I don't know of any special easier proof for the case $n=3$, specifically: ...
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1answer
23 views

Help with inequality: arithmetic vs weighted geometric

Let $p$, $q$ be positive real numbers such that $p+q < 1$. Prove that $$ \frac{p+q}{2} \leq \left( p^p q^q \right)^{1/(p+q)} $$ I'm not sure the assumption $p+q < 1$ is really necessary. ...
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1answer
39 views

Trying to prove an absolute value inequality $\left | a\sqrt{2} -b \right | > \frac{1}{2(a+b)}$

I am trying to prove that: $$\left | a\sqrt{2} -b \right | > \frac{1}{2(a+b)}$$ I was given that $a$ and $b$ are any positive integers. Can someone please help me? Thanks.
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1answer
46 views

Solving an easy inequality

Please, help me to solve the inequality: $$\sqrt{x^2+2x} > -3-x^2$$ I think I solved it, could you check it, guys?
1
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1answer
32 views

Help with an inequality involving a convex function

Let $a< f(x) < b $, $x \in \Omega $, $\mu(\Omega )=1 $, and set $t=\int f d \mu $. Then $a < t < b $. Suppose $\phi $ is a convex function on $(a,b) $ then by definition of convexity ...
2
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1answer
90 views

How prove $f(x)\le f(b)$. if $f(x)$ is continuous everywhere in [a,b], differentiable except at a countable number of points in [a,b]

QUestion: let $f(x)$ is continuous everywhere in [a,b], differentiable except at a countable number of points in [a,b].and $f'(x)\ge 0$ show that $$f(x)\le f(b)$$ This problem is from ...
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0answers
39 views

What are the most useful inequalities?

From a general point of view, when attacking problems where it could be useful, which are some of the most useful or handy inequalities that a mathematician can use as tools, based on your own ...
3
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2answers
38 views

show that $a{ \left\| x \right\| }_{ 1 }\le { \left\| x \right\| }_{ 2 }\le b{ \left\| x \right\| }_{ 1 }$

show that there exist positive numbers a,b such that $a{ \left\| x \right\| }_{ 1 }\le { \left\| x \right\| }_{ 2 }\le b{ \left\| x \right\| }_{ 1 } $ for all $x\in { R }^{ N }$ Find the largest ...
0
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0answers
31 views

A specific Inequality relating two quantities

Is $4a^3b-4ab^3-6b^2-2a^4<0$ always holds given $(a,b)\neq (0,0)$? Or else for what suitable constants can the inequality hold? Is there a more smarter way than using partial derivatives?
2
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2answers
52 views

General Advice on the proof of inequalities

I've got a pretty basic question that has been slightly confusing to me (I kinda understand this, but I need some affirmation). Basically, say I have an inequality $(2n+3)^2>4(n+1)(n+2)$, which, ...
4
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1answer
105 views

How prove The triangle inequality $\rho{(a,b)}\le \rho{(a,c)}+\rho{(c,b)}$

Question: let $a,b,c$ be complex numbers,and such $$|a|<1,|b|<1,|c|<1$$ let $$\rho{(x,y)}=\left|\dfrac{x-y}{1-\overline{x}y}\right|$$ show that $$\rho{(a,b)}\le ...
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0answers
27 views

Determine and sketch the pairs $(x,y)$ in $\mathbb{R} \times \mathbb{R}$ that satisfy some inequality

a) $|x| \leq |y|$ Continue my explanation below... If $y \geq 0$, then $-y \leq x \leq y$ and we get the region in the upper half-plane on or between the lines $y = x$ and $y = -x$
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3answers
101 views

An Inequality; $a^2+b^2=1$

$a,b$ are tho real numbers such that $a^2+b^2=1$. To prove that ; $$\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}+\dfrac{1}{ab+1}\geq\dfrac{3}{1+\cfrac{(a+b)^2}{4}}$$ When I first saw this question, I thought ...
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2answers
26 views

Modulus Inequality number of integer solution [closed]

Find the number of integer solutions to |x-1|+|y+1| < 5
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1answer
39 views

Solve $\ln(x^2 - 2x -2) \leq 0$

I'm trying to solve the inequality $\ln(x^2 - 2x -2) \leq 0$ Just want to make sure that I'm doing it right. $$\ln(x^2 - 2x -2) \leq 0$$ $x^2 - 2x -2 \leq e^0$ since $e^x$ is a strictly increasing ...
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3answers
87 views

How can we prove that $\left | (a-b)^{2}+ab \right | \geq \left | ab \right |$?

I have the multivariable function : $$f_{\alpha}(x,y)=\frac{\left | xy \right |^{\alpha}}{x^{2}-xy+y^2}$$ I think that an upper bound in $(0,0)$ for $\alpha > 1$ is : $$\left | f_{\alpha}(x,y) ...
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2answers
25 views

Meal Platters Optimization Problem

Mark has to buy hamburgers, hot dogs, and pig's feet for an event. The restaurant he is purchasing from offers two Platter options. Platter A comes with 4 hamburgers, 3 hot dogs, and 2 pig's feet. ...
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2answers
60 views

Why does $E(XY)^2 \le E(X^2)E(Y^2)$ fail to hold for complex $X, Y$?

Does cauchy schwartz inequality hold only for real vectors? Why is that so?
2
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1answer
79 views

For which values of $n$ does the following inequality stand?

For which values of $n$, does it stand that: $$-3n^4\log^2 n \geq -3 n^4 \sqrt{n} \quad ?$$
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0answers
57 views

Usefulness of $\frac{ac}{b}<a-b+c$

For $$a<b<0<c$$ I have a proof that shows that $$\frac{ac}{b}>a-b+c$$ But if $$a<0<b<c$$ Then $$\frac{ac}{b}<a-b+c$$ What I was wondering is how useful are these ...
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1answer
29 views

Conversion of a general linear program into a standard linear program

I am trying to teach myself the basics of optimization of linear programmes, for example the following question: How do I tackle such a question?
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0answers
35 views

Why is it true that $\forall b\in(0,1): (1-b)\left(e(1-b)\right)^{\frac{b}{1-b}}\geq\prod\limits_{n=2}^{\infty}n^{-b^n}\geq 0$

Why is it true that $$\forall b\in(0,1)$$ $$1\geq(1-b)\left(e(1-b)\right)^{\frac{b}{1-b}}\geq\prod\limits_{n=2}^{\infty}n^{-b^n}\geq 0$$ Note: Let $$f(x)=\prod\limits_{n=2}^{\infty}n^{-b^n}$$ Then ...
0
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0answers
25 views

About Robin's inequality and his work on counterexamples to the inequality.

Did Robin's work on his inequality and its relation to Riemann's Hypothesis prove that any counterexample to the inequality could not have a prime divisor with an exponent of 5 or more? If so how did ...
0
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1answer
57 views

Is this a Lipschitz continuous function?

Is $$\frac{xy}{1 + x^2 + y^2}$$ Lipschitz continuous on $x^2 + y^2 \le 4$? I've tried using Cauchy-Schwarz intequality but got nothing. I also tried to find out whether $xy$ is Lipschitz but failed ...
0
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3answers
59 views

Spivak's Calculus chapter 1 problem 12 v

I am having trouble proving $$|x|-|y|≤|x-y|.$$ In the solutions it says $$|x|=|y-(y-x)|≤|y|+|y-x|, \quad \text{so} \quad |x|-|y|≤|x-y|.$$ Am I missing something here? How did he get $|x-y|$ on the ...
1
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1answer
41 views

Show that inequality is correct for natural $n$

Show that the following inequality is correct for all natural $n$ : $$(2n+1)^n\geq(2n)^n+(2n-1)^n$$ I've tried throwing the $(2n-1)^n$ or $(2n)^n$ on the left side and using formula of subtraction ...
7
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1answer
113 views

Proving the natural log inequality $\frac{3}{4} < \frac{1}{\sqrt{3}} \log(2 + \sqrt{3}) < \frac{\pi}{4}$

I stumbled upon the following sharp inequality $$ \frac{3}{4} < \frac{1}{\sqrt{3}} \log(2 + \sqrt{3}) < \frac{\pi}{4} $$ I got a bit progress on the first part, but the second inequality ...