Questions on proving and manipulating inequalities.

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1
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2answers
18 views

How is using inequalities $\ge$ and $\le$ different for being used to solve equations?

I seriously want to know the difference finding the solution to equation and an inequality using $\ge$ and $\le$. I know how to solve inequalities involving variables like $9x+5$ $\ge$ $42$, but how ...
1
vote
4answers
29 views

prove that this equality is always right for each positive x and y.

prove that this inequality is hold for each positive x,y. $x\over\sqrt{y}$ + $y\over\sqrt{x}$ $\ge$ $\sqrt{x}$ + $\sqrt{y}$ I want a detailed way of solving the question.
0
votes
0answers
14 views

Find inequality for gaussian density

Let $C>0$ be a fixed constant. Is it true that $$Cx^2 e^{-x^2}\leq e^{-\frac{x^2}{C}}?$$ More generally, if we have a power $x^p$ in front of the exponential, do we have that $$(C^{1/2}x)^p\leq ...
0
votes
0answers
18 views

Problem in proving norm

I have a linear space V that includes the continuous functions from [-$\pi, \pi$] to the complex set C, of the form: $$f(x) = a\cos(t)+b\sin(t) $$ where $a$ and $b$ are complex numbers. I want to ...
4
votes
3answers
73 views

Prove the inequality$\int_0^{+\infty} {\sin x \over x}dx<\int_0^\pi {\sin x \over x}dx$

Prove: $$\int_0^{+\infty} {\sin x \over x}dx<\int_0^\pi {\sin x \over x}dx$$ Here is my answer,but I want a different way to prove it. \begin{aligned} \int_0^{+\infty} {\sin x \over ...
3
votes
2answers
54 views

An inequality with $a_n=\int_0^1 \frac{\mathrm{d}x}{\underbrace{\sqrt{2+\sqrt{2+\dots+\sqrt{2x}}}}_{n \text { times}}}$

Let the sequence $(a_n)_n$ defined by $$a_n=\int_0^1 \frac{\mathrm{d}x}{\underbrace{\sqrt{2+\sqrt{2+\dots+\sqrt{2x}}}}_{n \text { times}}}$$ 1)Prove that $$\frac12 \leq a_n \leq ...
2
votes
5answers
51 views

Prove that $e^x \ge$ its Maclaurin polynomial with n terms [on hold]

a) show that $e^x \geq 1+x$ for all $x\geq 0$ b) deduce that $e^x \geq 1+x+\frac{1}{2}x^2$ for $x\geq0$ c) use induction to prove that for $x\geq 0, n\in \mathbb{N}$ $$e^x\ge ...
1
vote
1answer
9 views

Find all of the exact solutions of the equation and then list those solutions which are in the interval [0, 2pi)

This is for trigonometric equations and inequalities: Find all of the exact solutions of the equation and then list those solutions which are in the interval [0, 2pi) Cos(9x)=9
-1
votes
2answers
46 views

Help me to prove this inequality. [on hold]

$$\text{Let } a,b\in R , a\neq 0. \text{ Show that } a^2+b^2+\frac{1}{a^2}+\frac{b}{a}\ge \sqrt{3}.$$
3
votes
1answer
27 views

Prelude to Cauchy-Schwarz, Quadratic proof.

I have a problem in trying to prove the following observation: "Show that if $ a,b,c \in \mathbb{R} $ are such that for all $ \lambda \in \mathbb{R} $, $a\lambda^2 + b\lambda +c \geq 0 $ then $ b^2 - ...
-1
votes
2answers
19 views

Prove that the image of $f: (0, \infty) \to R$ is contained in $[2, \infty)$. [on hold]

where $f(x) = x + 1/x$ Any help is appreciated, what I did was completely wrong haha..
0
votes
2answers
36 views

How to estimate $\sum_{n=0}^\infty (\log\log2)^n/n! $ from below?

How to show that the following inequality holds: $$\sum_{n=0}^\infty \frac{(\log\log2)^n}{n!}>\frac 35$$ Is it possible to prove this using induction?
4
votes
4answers
45 views

How to solve a convoluted absolute value inequality?

$$ \lvert \lvert x-2\rvert -3\rvert \lt 5 $$ How can I attack this the best way? I see that both sides are positive. Squaring yields: $$ \lvert x-2\rvert ^2 -6 \lvert x-2\rvert +9\lt 25 $$ $$ ...
-1
votes
1answer
26 views

An exponential inequality

Assume $a(t)\geq 0$ and $b(t)\geq 0$. i can show the following inequality $\mid e^{-\int_0^ta(s)ds}-e^{-\int_0^tb(s)ds} \mid\leq T\max_{0\leq t \leq T}\mid a(t)-b(t)\mid$ by writing $\mid ...
7
votes
2answers
75 views

minimize a function using AM-GM inequality

I want to minimize the function $$ \frac{x}{1-x^2} + \frac{y}{1-y^2} + \frac{z}{1-z^2} $$ subject to the constraint $$x^2 + y^2 + z^2 = 1 \space\text{and} \space x,y,z > 0$$ Wolfram Alpha tells ...
5
votes
3answers
121 views

Proof of Nesbitt's Inequality?

I just thought of this proof but I can't seem to get it to work. Let $a,b,c>0$, prove that $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge \frac{3}{2}$$ Proof: Since the inequality is homogeneous, ...
2
votes
2answers
60 views

Proving $|P(A\cap B)-P(A)P(B)|\leq \frac{1}4$

Let $A$ and $B$ be two events of a probability space. Prove that $\displaystyle|P(A\cap B)-P(A)P(B)|\leq \frac{1}4$ I think it's a very challenging problem, and I've made no progress so far ... ...
4
votes
1answer
33 views

How to prove this inequality relating to trigonometric function?

In a triangle, A, B, C are three corners of the triangle, try to prove that : $$\root 3 \of {1 - \sin A\sin B} + \root 3 \of {1 - \sin B\sin C} + \root 3 \of {1 - \sin C\sin A} \geqslant {3 \over ...
4
votes
5answers
112 views

Showing that $e^{-2} < \ln 2$

I have to prove the following inequality: $e^{-2} < \ln2.$ Using Bernoulli's inequality, I showed that $2 \leq e$, and using this result I tried to simplify the inequality by using an upper ...
5
votes
3answers
253 views

Beautiful cyclic inequality

Prove that cyclic sum of $\displaystyle \sum_{\text{cyclic}} \dfrac{a^3}{a^2+ab+b^2} \geq \dfrac{a+b+c}{3}$ , if $a, b, c > 0$ I'm really stuck on this one. Tried some stuff involving QM> ...
1
vote
3answers
44 views

Prove that: $\sqrt[3]{a+2b}+\sqrt[3]{b+2c}+\sqrt[3]{c+2a}\le 3\sqrt[3]{3}$

Given $a,b,c>0$ and $a+b+c=3$. Prove that: $\sqrt[3]{a+2b}+\sqrt[3]{b+2c}+\sqrt[3]{c+2a}\le 3\sqrt[3]{3}$
-3
votes
3answers
65 views

Prove that: $\sqrt{4-a^2}+\sqrt{4-b^2}+\sqrt{4-c^2}\le 3\sqrt{3}$ [on hold]

if $0<a,b,c\le2$ . Prove that: $\sqrt{4-a^2}+\sqrt{4-b^2}+\sqrt{4-c^2}\le 3\sqrt{3}$
0
votes
0answers
19 views

Machine Floating Point Theorem

Completely stuck on this floating point question. Let $x \in \mathbb{R}$ have the following floating point representation: $$ x = (-1)^s[0.a_1a_2\dots a_ta_{t+1}\dots]\cdot \beta^e $$ [Where $\beta$ ...
0
votes
2answers
20 views

Inequalities with expected value on one side and probability on the other

In a part of a proof I am following, the author states that $$\displaystyle \mathbb{E}\left[\frac{|X_n - X|}{1 + |X_n - X|}\right] \leq \epsilon + \mathbb{P}(|X_n - X| > \epsilon)$$ and ...
2
votes
1answer
27 views

Inequalities with more than one absolute value

I saw a question which asked to find all the solutions to: $|x+2|+|x-5|=7$ For $x\leq -2$, the answer is $-2$. For $-2< x <5$, the answer is $R$. For $x>5$, the answer is $5$. First I ...
3
votes
3answers
50 views

Logarithmic inequality for a>1

Is $\log_{\sqrt a}(a+1)+\log_{a+1}\sqrt a\ge \sqrt6$ always true for $a>1$? What is the approach? Do we check the first a's and then form a induction hypothesis?
3
votes
2answers
40 views

Show that if x,y are and $ x^4y^2+x^2+2x^3y+6x^2y+8 \leq 0 $ then $x \geq -1/6 $

Show that if x,y are real and $ x^4y^2+x^2+2x^3y+6x^2y+8 \leq 0 $ then $x \geq -1/6 $ So far I've tried factoring $x^2$ and throwing the 8 on the LHS, but can't get to the needed result. Help would ...
0
votes
1answer
27 views

prove length-like function is convex

I'm trying to prove that $ F(u)= \int\limits_0^1 (1+(du/dx)^2)^{1/2}$ is a convex function of u ; however after squaring both side twice of $(1+(d(tu_1)/dx)^2)^{1/2}+(1+(d((1-t)u_2)/dx)^2)^{1/2} ...
1
vote
1answer
18 views

a problem about a convex function

I'm trying to prove that $ (1+x^2)^ {1/2} $ is a convex function; however after squaring both side twice of $ (1+(tx)^2)^ {1/2} + (1+((1-t)y)^2)^ {1/2} > (1+(tx+(1-t)y)^2)^ {1/2}$ I got so many ...
2
votes
2answers
42 views

For every integer $n \geq 1$, prove that $3^n \geq n^2$.

It's been a while since I've done induction, and I feel like I'm missing something really simple. What I have is this: Base Case: $n=1$ $$3^n \geq n^2 \implies 3 \geq 1$$ Inductive ...
0
votes
3answers
39 views

How to solve this inequality? I tried a lot of times, but answer is not correct.

My answer is [4;9], but it must be [0;9], I don't understand what's wrong. Could you give me solution?
2
votes
2answers
45 views

Show that $2^{n-1}(x^n + y^n ) \geqslant (x+y)^n $

Let $x,y$ be real numbers such that $x+y\geqslant 0$. Prove that $2^{n-1}(x^n + y^n ) \geqslant (x+y)^n $ for every $n \in N$. Can we prove this using only inequalities,without using induction?
1
vote
1answer
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} ...
-1
votes
3answers
46 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 ...
1
vote
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
vote
1answer
84 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 ...
8
votes
5answers
145 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} ...
-1
votes
1answer
44 views

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

Could you explain to me why $10+2p$ is divided by $p$ in this solution?
0
votes
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 ...
0
votes
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)
1
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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 $(-2, 2)$
2
votes
2answers
85 views

An inequality, which is supposed to be simple

Let $x,y,z\in\mathbb{R}$.Let $xy+yz+xz=1$. Prove:$\displaystyle \frac{x}{\sqrt{x^2+1}}+\frac{y}{\sqrt{y^2+1}}+\frac{z}{\sqrt{z^2+1}}\leq \frac{3}{2}$
3
votes
0answers
19 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
votes
4answers
81 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 \ ...
10
votes
4answers
160 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
votes
1answer
23 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, ...
0
votes
3answers
46 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
votes
2answers
19 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
0
votes
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 $$x^2 -4x-12 < 0$$ ...
0
votes
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 ...