Questions on proving and manipulating inequalities.

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4answers
48 views

Solve the inequality: $|x^2 − 4| < 2$

This is a question on a calculus assignment our class received, I am a little confused on a few parts to the solution, can someone clear a few things up with it? Since $x^2-4 = 0$ that means $x = 2$ ...
1
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2answers
17 views

Is the following inequality valid or not?

$ 1 > 1 - \dfrac{1}{x}$ for every $x \in \mathbb{N} $. Obviously the limit of $1-\dfrac{1}{x}$ as x approaches infinity is $1$, but is this inequality 100% valid? Does it need a proof, and if so, ...
1
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3answers
42 views

prove inequality

I have to show: $\displaystyle\frac{1}{1^4}+\frac{1}{2^4}+...+\frac{1}{n^4} \le 2-\frac{1}{\sqrt{n}}$ for natural $n$ I tried to show it by induction (but I think it could be possible to show it ...
2
votes
3answers
45 views

Showing that $x+ cos x - 1 > 0$ for all $x > 0$

I got this problem: Show that for all $0<x$, $0<x+cos x - 1$ I tried to show it several times but none worked. I showed that $lim_{x\to\infty} (x+cos x - 1) = \infty$ by using the Squeeze ...
2
votes
1answer
100 views

symetric inequality for a rational function of three variables

$a,b,c$ are positive real numbers such that $a+b+c=1$. Prove that: $$\dfrac{a^3}{a^2+b^2}+\dfrac{b^3}{b^2+c^2}+\dfrac{c^3}{c^2+a^2} \geqslant \dfrac{1}{2} $$ I have tried with Cauchy-Schwarz ...
1
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0answers
22 views

Find condition on $A$ and $B$ so that $|q_o|+|q_1|<1$.

I have to find condition on $A$ and $B$ so that $$|q_o|+|q_1|<1$$ Where $$q_0=\frac{-1-4A+C+(2B+1)\sqrt{8A+(B+C-1)^2}-B(2B+2C+1)}{8A-4(B+C)}$$ and ...
1
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1answer
29 views

Inequality regarding exponential function

For every positive $x$ and for every $n$ show that $(1+\sqrt{\frac{x}{n}}+\sqrt[3]{\frac{x}{n}})e^{n\arctan{\frac{x}{n}}}>e^x$. I plotted it, it seems that the inequality holds. Any ideas how to ...
5
votes
3answers
113 views

Inequality $\binom{2n}{n}\leq 4^n$

I would like to prove the following inequality, for $n=0,1,2,...$, $$ \binom{2n}{n}\leq 4^n.$$ I already proved it by induction, and I'm looking for another proof.
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3answers
40 views

Using the comparison test to evaluate $\int_1^\infty\frac{1}{1+x^2+16x^4}dx$?

So using the comparison test to evaluate $\int_1^\infty\frac{1}{1+x^2+16x^4}dx$, and we're given $\int_1^\infty\frac{1}{4x^2}dx$. So I have been trying to set up an inequality to use, but I can't seem ...
0
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1answer
39 views

Verification of proof that $x^p+y^p<(x+y)^p$

I came across the inequality $x^p+y^p<(x+y)^p$, $x,y \in \mathbb{R}^+$, $p>1$ during working on an assignment. While I think I have a proof down, I'm not 100% confident about the steps taken. ...
1
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1answer
31 views

If$ x^2 + y^2 + Ax + By + C = 0 .$ Find the condition on $A, B$ and $C$ such that this represents the equation of a circle.

Also find the center and radius of the circle Here's my solution, I'm not sure if it's correct or not (specifically the conditions on $A$, $B$ and $C$. I feel that my conditioning is invalid and that ...
4
votes
3answers
54 views

Show that the solution of an initial value problem is always less than a given constant

My try is that $$\frac{dy}{dt} =(y-3)e^{\cos ty}$$ $$\frac{dy}{y-3}= e^{\cos ty}dt$$ $$\ln (y-3)=-\frac{e^{\cos ty}}{\sin ty} +c$$ my steps is correct or I made mistakes ? please help to solve ...
0
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2answers
53 views

How prove this integral inequality $1\le \int_{0}^{2}f(x)\,dx\le 3$

Let $f(x)$ be differentiable on $[0,2]$,and such $f(0)=f(2)=1,|f'(x)|\le 1,x\in[0,2]$. Show that $$1\le \int_{0}^{2}f(x)\,dx\le 3.$$ I want use Taylor theorem ...
1
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1answer
49 views

Symmetric inequality for a rational function of three variables

If $x,y,z$ are positive real numbers such that $xyz \geqslant 1$ prove: $$\dfrac{x^3+y^3}{x^2+xy+y^2}+\dfrac{y^3+z^3}{y^2+yz+z^2}+\dfrac{x^3+z^3}{x^2+xz+z^2} \geqslant 2$$ I have tried with ...
2
votes
1answer
22 views

Supremum of sum of two sequences: $\sup (x_n+y_n) \le \sup x_n + \sup y_n$

Prove that $\sup\{x_n+y_n\}\leq \sup\{x_n\}+\sup\{y_n\}$, if both sups are finite. Furthermore, prove that $\limsup\{x_n+y_n\}\leq \limsup\{x_n\}+\limsup\{y_n\}$ if both limsups are finite.
4
votes
4answers
69 views

How to prove this inequality with $a+b+c=1$

Question: For any $a,b,c\in \mathbb{R}$ such that $a+b+c=1$ and $abc>0$, show that $$ab+bc+ac<\dfrac{\sqrt{abc}}{2}+\dfrac{1}{4}.$$ My idea: let $$a+b+c=p=1, \quad ab+bc+ac=q,\quad ...
-1
votes
1answer
29 views

Inequality of expectation of random variables [on hold]

Let $X$ and $Y$ be two continuous random variables in $\mathbb{R}$. $X$ and $Y$ may not have any kind of stochastic dominance properties. Let $f$ be a strictly monotonically increasing concave ...
0
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1answer
57 views

$x^2+y^2+z^2 +3 \geq 2(xy+xz+yz)$, for $xyz=1$ [on hold]

$x,y,z > 0$ such that $xyz=1$ can you prove that $x^2+y^2+z^2 +3 \geq 2(xy+xz+yz)$ without lagrange multiplier Thanks
1
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1answer
21 views

Prove for relatively prime numbers.

Prove that for relatively prime positive integers $a$ and $b$, the equation $ax+by=c$ must have non-negative integer solution if $c>ab-a-b$.
2
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2answers
52 views

Find the smallest $a>1$ such that $\frac{a+\sin x}{a+\sin y} \leq e^{(y-x)}$ for all $x \leq y$

Can anyone please help me with the following question: Find the smallest $a>1$ such that $$\frac{a+\sin x}{a+\sin y} \leq e^{(y-x)}$$ for all $x \leq y$ My attempt: I think we should rearrange ...
1
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2answers
51 views

How to write a formal proof of the statement: if $x<3$ then $10-2x>4$?

Prove: For all real numbers $x$, if $x<3$ then $10-2x>4$. Proof: Let $x \in \mathbb{R}$, such that $x<3$. We have the following sequence of implications: $10-2x>4 \Rightarrow -2x>4 ...
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1answer
26 views

Induction Proof of Inequality Involving Summation [on hold]

I really need help with the following exercise! Show with induction that $\sum _{k=1}^n\left(k^2\right)<n^3$, for $n>1$ .
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3answers
34 views

Limit-related inequalities with absolute values

Recently I decided to learn calculus on my own and I stumbled across something which I cannot figure why is correct. Let $f$ be some function for which you know only that if $0<|x-3|<1$, then ...
1
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2answers
50 views

minimal value of $x^2+2y^2+5z^2$ with constraint.

$x,y,z>0$, and $xy+yz+zx=1$. I need to find the minimum value of $x^2+2y^2+5z^2$ In general what can we say about the minimal value of $\frac{ax^2+by^2+cz^2}{xy+xz+yz}$, over all positive numbers ...
0
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0answers
26 views

Characterize the set that solves the Diophantine inequality

Given that $a, b, c$ are integers and $2\max(|b|, |3a + b|) \le \min(|c|, |3a+2b+c|)$ What characterizes the solution set in $\mathbb{Z}^3$? Obviously this is equivalent to the system of linear ...
0
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0answers
18 views

question on inequalities

Consider the following inequality: b + g(1-2b) > 0 which of the following two is correct? b > -g(1-2b) => b > g(2b-1) => g < b/(2b-1) or g(1-2b) > -b => g > -b/(1-2b) => g > b/(2b-1) ...
0
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1answer
57 views

prove this inequality related to probability and information theory

How do I prove this? I'm thinking I should use Jensen's inequality somehow. $$\sum_K p_k(1-p_k) \le -\sum_K p_k\log p_k$$ The assumption that $\sum_K p_k=1$ holds.
1
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4answers
86 views

How to show that $a+b> \sqrt{a^2+b^2-ab}, \qquad a, b >0$

How do you show that $$a+b> \sqrt{a^2+b^2-ab}, \qquad a, b >0$$ I could write $\sqrt{a^2+b^2-ab}=\sqrt{(a+b)^2-3ab}$, but this seems to lead nowhere.
2
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0answers
60 views

An entropy inequality

Let $f:[0,2\pi]\to \mathbb{R}$ be a smooth, positive function such that $f(0)=f(2\pi)$, and $\int_0^{2\pi}fd\theta=2\pi.$ Is it true that $$2\int_0^{2\pi}f\ln fd\theta- 2\int_0^{2\pi}\ln ...
2
votes
3answers
69 views

Maximum value of $a+b$ given that $\frac{1}{a} + \frac{1}{b} = \frac{1}{20}$

What is the maximum value of $a+b$ given that $\frac{1}{a} + \frac{1}{b} = \frac{1}{20}$ here $a,b \in \mathbb{Z^+}$? What I have gotten so far: From the above, $\frac{a+b}{ab} = ...
2
votes
1answer
20 views

An inequality using Sobolev norms

Let $\| \cdot \|_{H^s(\mathbb R)}$ be the usual Sobolev norm in $\mathbb R$ and $r>0$. If we have $$ \|f\|_{L^\infty(\mathbb R)} \le \| f\|_{H^k(\mathbb R)} $$ for all $k>r$, then the inequality ...
0
votes
2answers
56 views

Solve $|x-2| \leq 2|x|$

This is an in-class example we were given in calculus class, I am having some difficulty understanding one of the instructor's steps. The following is my attempt of the question: Since this is an ...
4
votes
2answers
74 views

Inequality $\frac{\sqrt a+\sqrt b+\sqrt c}{2}\ge\frac{1}{\sqrt a}+\frac{1}{\sqrt b}+\frac{1}{\sqrt c}$ with weird condition

I want to prove the following inequality: $$\frac{\sqrt a+\sqrt b+\sqrt c}{2}\ge\frac{1}{\sqrt a}+\frac{1}{\sqrt b}+\frac{1}{\sqrt c}$$ Where $a,b,c$ are positive reals and with the horrible ...
0
votes
1answer
43 views

Inequality with square roots

I need help with another inequality. $\sqrt{2x+4}-2\sqrt{2-x}>\dfrac{12x-8}{\sqrt{9x^2+16}}$. Squaring leads nowhere as you get a polynomial of degree 4 with no rational roots.
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2answers
59 views

Writing a proof of an inequality between fractions

I have no idea how to do this. Suppose $x,y,z,n$ are positive integers. Given that $\frac{x}{y} < \frac{z}{n}$, prove that $$\frac{x}{y} < \frac{x+z}{y+n} $$
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1answer
37 views

Find the greatest value of this expression

Let $x$, $y$, $z$ be positive real numbers with $x + y + z = 3$. Find the greatest value of the expression \begin{equation} P = \sqrt{(x+1)(y^2 + 2)(z^3 + 3)} + \sqrt{(y+1)(z^2 + 2)(x^3 + 3)} + ...
3
votes
3answers
84 views

Prove that $x+\frac{1}{2x}-\frac{1}{8x^3}<\sqrt{x^2+1}<x+\frac{1}{2x}$ for all positive real numbers x

This is a math problem from the German Math Olympiad, but in this case I do not know where to start, probably because I do not have enough intuition regarding inequalities. However, I tried to apply ...
0
votes
1answer
42 views

inequalities and their solutions

a. Since: 9|x + 9| + 6 > 5 9|x + 9| > -1 |x + 9| > -1/9 => always true, every real x value is a solution so it has infinitely many solutions. (−∞, ∞) How would I graph this solution set on a ...
1
vote
1answer
35 views

Inequalities with function $e^{x^2+e^{x^2}}$

Let $f(x)=e^{x^2+e^{x^2}}$ for $x\in\mathbb{R}$. How to prove that for any $a,b>0$, $a\neq b$ the following inequalites hold $$(b-a)f(\frac{a+b}{2}) < \int_a^b f(x)\ dx < ...
6
votes
1answer
387 views

Inequality involving square roots

I need help with this inequality: $\sqrt x +\sqrt{x+7} + 2\sqrt{x^2+7x} <35-2x$ It doesn't seem solvable. All roots of the corresponding equation are irrational.
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0answers
17 views

How to solve ordinary differential inequations with vector variables?

Given $a\in\mathcal{R}_+^d$ and $s\in\mathcal{R}^d$,we wanna a function f(.) which maps s to a vector $f=\begin{bmatrix}f(s_1),\cdots,f(s_d)\end{bmatrix}^T$ and satisify the following inequation. ...
1
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2answers
38 views

Turning $2\le x$ into $\sqrt{1+\frac{4}{x^6}}\le \sqrt{5}$?

I am supposed to turn $2\le x$ into $\sqrt{1+\frac{4}{x^6}}\le \sqrt{5}$, and I have no idea on how to approach this. I'll post my steps, even though I don't think they'll be of much help. $$2\le x ...
3
votes
1answer
53 views

Does such a function exist always?

Suppose that $f(x)$ is some smooth function on $[0,1]$ with $f(x) \geq c > 0$. Can we always find a function $g(x)$ smooth satisfying $g'(x) \not= 0$ for all $x \in [0,1]$ and $f'(x)g'(x) + ...
0
votes
1answer
19 views

Prove that $ \max(f(n),g(n)) \le f(n)+g(n) \le 2\max(f(n),g(n)) $ [closed]

Particularly, can someone show that $f(n)+g(n) \le 2\max(f(n),g(n))$ is true for $f,g$ nonnegative?
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votes
3answers
56 views

Prove $(wx+yz)^2 \leq (w^2 + y^2)(x^2+z^2)$ using order axioms [closed]

Problem: Using order axioms, prove that for $\forall w,x,y,z \in \mathbb{R}$ $$(wx+yz)^2 \leq (w^2 + y^2)(x^2+z^2)$$ Progress. I've tried to foil out both sides, resulting in $2wxyz ≤ x^2y^2 + ...
3
votes
1answer
66 views

Is this proof that $\lfloor x \rfloor \geq n \left\lfloor \frac{x}{n} \right\rfloor$ correct?

In this text the fractional part of a real $x$ shall be denoted $\{x\}$, such that $x = \lfloor x \rfloor + \{x\}$. Theorem: $$ \forall x \in \mathbb{R}_{\geq 0} \forall n \in \mathbb{N}_{\geq 1} : ...
3
votes
2answers
47 views

How to show that $\|a+b+c\|^2\leq 3\|a\|^2+3\|b\|^2+3\|c\|^2$

Show that $$\|a+b+c\|^2\leq 3\|a\|^2+3\|b\|^2+3\|c\|^2$$ where $a,b,c$ are in some Hilbert space $(H,\langle\cdot,\cdot \rangle)$? I see that we have $$\|a+b\|^2\leq2\|a\|^2 +2 \|b\|^2$$ due to the ...
3
votes
3answers
46 views

Prove or Disprove Inequality By Induction

Prove or Disprove $\sum_{i=0}^n(2i)^3 \le (8n)^3 $ If true, prove using induction. If false, give the smallest value of n that is a counter example and the values for the left and right hand sides ...
0
votes
5answers
80 views

Prove or give a counterexample: For all $x > 0$, $x^2 + 1 < (x+1)^2 \le 2(x^2 + 1)$

I am working on the following problem from Lay's Analysis with an Introduction to Proof: Prove or give a counter example: For all $x > 0$ we have $x^2 + 1 < (x+1)^2 \le 2(x^2 + 1)$ Now, ...
1
vote
2answers
63 views

How to prove that $0 < 1$ from the order axioms of $\mathbb R$?

My homework question: From the order axioms for $\mathbb{R}$, show that $0 < 1$. [Hint: From the field axioms, $0 \not=1$. By the trichotomy property, either $0<1$ or $4<0$. Assuming $1 ...