Questions on proving and manipulating inequalities.

learn more… | top users | synonyms (1)

-2
votes
4answers
46 views

max or min value of a function : y = x + 16 /(x+3) where x > 0, without using differentiation [closed]

Many thanks for your help. I don't know how to solve it. also, How do it determine it is a minimum or maximum. Noted. The student does not know about differentiation.
-2
votes
2answers
46 views

Given $a,b,c$ belongs to $R^{+}$, prove that $(a^2+1)(b^2+1)(c^2+1) \geq 8abc$ [closed]

Kindly for your help. Thanks in advance.
0
votes
0answers
48 views

Differential Calculus for exponential growth

Let the function $f(z)=\frac{z}{\sqrt{1+|z|^2}}(e^{-1+\sqrt{1+|z|^2}}-\sqrt{1+|z|^2})$ for all $z\in \mathbb C$. I want to prove that for all $\varepsilon >0$ there exist a positive constant ...
3
votes
2answers
94 views

Prove that $a^7 + b^7 + c^7 \geq a^4b^3 + b^4c^3 + c^4a^3$

Prove that $a^7 + b^7 + c^7 \geq a^4b^3 + b^4c^3 + c^4a^3$ Values $a,b,c$ are all positive reals. I tried Muirhead and a few AM $\geq$ GM. This problem is equivalent to proving $a^4b^3 + b^4c^3 ...
1
vote
1answer
32 views

Proving an inequality involving integer polynomial

So we've got an integer polynomial $P$, and all we know about it is that $P(1) = 1$, $P(2) = 2$, and also $P(100) = -k$, where $k \in \mathbb{Z},\, k \geqslant 0$ - some unknown constant, which will ...
0
votes
1answer
45 views

Interpolation between derivatives

I am trying to prove the following: Let $u \in W^{2,2}(\mathbb R)$. Then $\| u^\prime \|_{L^2}^2 \leq \| u \|_{L^2} \| u^{\prime \prime} \|_{L^2}$ holds (these are meant to be weak derivatives). ...
3
votes
2answers
45 views

Prove the inequality $x_1^{y_1}+x_2^{y_2}+\cdots+x_n^{y_n} \geq x_1^{y_{\pi(1)}}+x_2^{y_{\pi(2)}}+\cdots+x_n^{y_{\pi(n)}}.$

Let $1 \leq x_1 \leq x_2 \leq \cdots \leq x_n$ and $1 \leq y_1 \leq y_2 \leq \cdots \leq y_n$. For any permutation $\pi$ prove the inequality $$ x_1^{y_1}+x_2^{y_2}+\cdots+x_n^{y_n} \geq ...
3
votes
1answer
38 views

Show inequality holds

Let $z \geq y \geq 1$. Show that $$ \sqrt{\frac{y}{1+z}} + \sqrt{\frac{z}{1+y}} + \sqrt{\frac{1}{y+z}} > 2 $$ These are actually the last steps of a bigger inequality, but I can't think of a nice ...
2
votes
2answers
141 views

subtracting inequalities if difference is positive

$$ x \geq y $$ $$ a \geq b $$ $$x+a \geq y+b $$ is valid but $$ x-a \geq y-b $$ is not valid Can we say the latter is valid if $x-a \geq 0$ ? Is it a proof or am I wrong? Are there counter ...
2
votes
0answers
62 views

When does equality hold in this inequality?

The following inequality can be proven as follows: Let $n\geq3$ and $0=a_0<a_1<\dots<a_{n+1}$ such that $a_1a_2+a_2a_3+\dots+a_{n-1}a_n=a_na_{n+1}$. Show that \begin{equation*} ...
0
votes
2answers
33 views

Proving exponential inequalities

I'm currently revising for an upcoming exam and am stuck on the following question. I have completed a similar question that involved cos and the mean value theorem I used the triangle inequality too, ...
0
votes
1answer
17 views

Bounds on functions via its derivatives

Suppose, we have a function $f$ where $f$ is: Contionuos. Non-Negative Has a derivative given by $f'$. Can we have a bound on $f$ in terms of its derivative $f'$? That is have an inequality that ...
1
vote
1answer
35 views

Using the triangle inequality to bound $\frac{x^3 + 3x + 1}{10-x^3}$ for $|x+1|<2$

How do I use the triangle inequality to bound the function $$f(x) = \frac{x^3 + 3x + 1}{10 - x^3}$$ on the interval $|x+1|<2$? I understand how the triangle inequality works, but using fractions ...
3
votes
0answers
43 views

Prove $\frac{1}{\pi^2}\int_0^x \left(\sin \pi t\right)^2\left[\frac{1}{(t-a)^2}+\frac{1}{(t+a)^2}\right]dt\geq \frac{x-a}{1+x-a}$

Prove that $$\frac{1}{\pi^2}\int_0^x \left(\sin \pi t\right)^2\left[\frac{1}{(t-a)^2}+\frac{1}{(t+a)^2}\right]dt\geq \frac{x-a}{1+x-a}$$ for every $x\geq a>0$. I do not know where to start! Any ...
5
votes
3answers
166 views

How to solve this inequality, with the hypothesis more complicated than the conclusion?

Given $x,y,z \in \mathbb{R}$ and $x,y,z>2,$ I want to show that if, $$\frac{1}{x^2-4}+\frac{1}{y^2-4}+\frac{1}{z^2-4} = \frac{1}{7}$$ then, $$\frac{1}{x+2} + \frac{1}{y+2} + \frac{1}{z+2} \leq ...
2
votes
5answers
103 views

inequalities with fraction problem x

$\frac{1}{x} > \frac{2x} {x^2 +2}$ solving this inequalities: My long solution (wrong) : multiplying $(x^2 + 2)^2 (x)^2 \dots$ (multiplying square of each denominator, getting rid of the > or ...
2
votes
0answers
15 views

Confused about Landau-notation and inequality

Let $f$ be a real valued function and $|f(x)| \le x^2\cdot C + o(x^3)$ as $x\to 0$, where $C \ge 0$ is a constant independent of x. Is it true that there is a $x_0$ such that for all $x\in [0,x_0]$ ...
5
votes
2answers
46 views

Metric in $\mathbb{P}_2$

I have to prove that $\mathbb{P}_2$ with the function $\delta(P,Q)$ defined by "Sine of the angle between two vector in $\mathbb{R}^3$ such that they correspond respectively to P and Q" is effectively ...
0
votes
0answers
32 views

inequality on Mean Value theorem

Good Day! I understood that By mean Value theorem: $$2 - t^{n-1} (1+t) = (1 - t)[θ^{n – 1} + (n - 1) θ^{n – 2} (1 + θ)]$$ where $$t < θ < 1$$ but How can I prove that: $$(1 - t)[θ^{n – 1} + ...
-1
votes
5answers
105 views

The interval in which $ab+bc+ca$ lies if $a^2+b^2+c^2=1$

What is the interval in which $$ab+bc+ca$$ lies if $$a^2+b^2+c^2=1$$.I have considered using the AM-GM inequality. But that's not working. Please provide some advice.
1
vote
1answer
64 views

Showing that $(a^2-a+1)(b^2-b+1)(c^2-c+1) \leq 7$

How can I show that $(a^2-a+1)(b^2-b+1)(c^2-c+1) \leq 7$ given that $a+b+c = 3?$ Attempt: Setting $x=2a-1, y=2b-1, z=2c-1,$ we obtain that $[(2a-1)^2+3][(2b-1)^2+3][(2c-1)^2+3] \leq 448,$ $s=x+y+z ...
3
votes
5answers
72 views

Prove that $(1+a)^x>1+ax$ when $x>1$ and $0<a<1$

Prove that $(1+a)^x>1+ax$ when $x>1$ and $0<a<1$ and $(1+a)^x<1+ax$ when $0<x<1$ and $0<a<1$ I was trying to do it the usual way which is to consider the function ...
0
votes
0answers
37 views

Pointwise inequality in $n$ dimensions

Suppose we have the system $h(x)=0$ and $f(x)=0$, where $h, f$ are real, decreasing and for which their first derivatives are negative (i.e. $\frac{\partial h}{\partial x} < 0$, $\frac{\partial ...
0
votes
2answers
33 views

A simple probability inequality

I am looking for a simple proof of the following inequality. Suppose that $X$ and $Y$ are two random variables. For any $x>0$ and $y>0$, $$ \Pr\{|X+Y|\ge x\} \ge\Pr\{|X|\ge x+y,|Y|\le y\}. ...
1
vote
2answers
63 views

solving inequalities with fractions on both sides

Solve this inequality: $(x^2 -2)/2 < (6x^2 -8x - 1)/(x+5)$. My solution: Multiply $(x+5)^2$ on both sides: $(x^2 -2)(x+5)^2 /2 = (x^4 + 10x^3 + 23x^2 -20x -50)/2$ $(6x^2 -8x - 1)(x+5) = 6x^3 ...
0
votes
4answers
59 views

solving a power 4 inequality

Solve this inequality: $x^4 - 3x^3 - 4x^2 \leq 0$ My answer (wrong) : $x^2 (x-4)(x+1)\leq0$ Case 1 $x^2 \geq 0$ , $x\geq4$ , $x\leq-1$ NO Case 2 $x^2 \geq 0$, $x\leq4$ , $x \geq -1$ ... so ...
2
votes
3answers
45 views

Solving a cubic inequality

$4x^3 - 3x^2 + x \ge 0$ , solve the inequality: My solution for this question: $x(4x^2-3x+1) \ge 0 $ $4x^2 - 3x + 1$ is not solvable, assume $x \in \mathbb R$ So $x \ge 0$ and $x \in \mathbb R$ ...
1
vote
1answer
54 views

I have a question on how to prove an inequality holds true.

How do I prove that for $a < b< c$ and $n \geq 3$ then the inequality $$b^n+c^n > a^n$$ holds true. I tried setting $c =b+n$ but I don't know we're to go from here !
0
votes
3answers
44 views

Solving this Inequation

Show that $e^x \geq x^e$ for $ 0 \lt x \lt \infty $. I tried to apply the normal logarithm here, which yields $x \geq e\times \ln(x)$ Still, I am kind of stuck here, anyone mind giving me a hand?
4
votes
6answers
539 views

How can I compare numbers raised to a square root

For example: $3^\sqrt5$ versus $5^\sqrt3$ I tried to write numbers as this: $3^{5^{\frac{1}{2}}}$ and then as $3^{\frac{1}{2}^5}$ But this method gives the wrong answer because $a^{(b^c)} \ne ...
0
votes
0answers
26 views

Decreasing of power function.

Show that $\frac{3}{4}(x-2)x^{-\frac{5}{2}}-(x+1)^{-\frac{3}{2}}<0$, wherer $x\ge0$. I tried by taking differentiation but then expression become more complicated. I also tried by checking the ...
3
votes
3answers
113 views

Absolute inequality derivation

I have been trying to prove an inequality that I am not even sure if it is even true or not. However I am experiencing great difficulties with this proof. I have an intuition that it is true and have ...
1
vote
3answers
67 views

How do I solve this rational inequality?

I'm having difficulty finding any help online for this equation: $$\frac{3x^2+2x-5}{2x^2+5x}\le0$$ I need to solve the inequality, and then put it in interval notation. So far I have taken the ...
1
vote
1answer
43 views

Chernoff-like bound for small intervals in tail distribution

I am searching for a Chernoff-like bound that controls the probability of small intervals in the tail distribution. More specifically, let $X_1, \ldots, X_n$ be independent random variables with ...
2
votes
3answers
62 views

$3x + 2 > 8$ solved not using order of operation?

So im basically relearning algebra, using a site to teach myself the basics. I read about bodmas then shortly after about inequalities. In the practice questions i got a bunch of questions where the ...
0
votes
1answer
86 views

How to prove the inequality $ (1+a+ab)(1+b+bc)(1+c+ca) \leq (1+a+a^2)(1+b+b^2)(1+c+c^2)?$

For $a,b,c>0$ prove the inequality $$ (1+a+ab)(1+b+bc)(1+c+ca) \leq (1+a+a^2)(1+b+b^2)(1+c+c^2). $$ I know that I should use the multiplicative rearrangement inequality but I am not sure how ...
1
vote
2answers
42 views

Why if $B = \{x : |x+1| ≤ 3 \}$ then $B$ equals $[ -4, \infty )$?

I really don't understand why $B$ is from $-4$ to infinity because $x+1 ≤ 3$ $x ≤ 2$ and $-3 ≤ x+1$ $-4 ≤ x$ Shouldn't it be $B = [-4, 2]$?
-1
votes
0answers
30 views

Inequalities involving ordered variables

For given real numbers $\lambda_1\geq\lambda_2\geq\lambda_3\geq\lambda_4\geq\lambda_5\geq\lambda_6$ and the set identity $\{i,j,k,l,m,n\}=\{1,2,3,4,5,6\}$, show that the inequality \begin{equation} ...
-2
votes
4answers
67 views

A.M.>G.M. of four numbers [duplicate]

Prove that arithmetic mean of 4 numbers is greater than geometric mean of the same 4 numbers, i.e. prove that $$\dfrac{a+b+c+d}{4} > (abcd)^{\frac1{4}}$$
5
votes
5answers
185 views

A unusual inequality about function $\ln$

These day,I met a unusual inequality when I solve a difficult problem, and proving the inequality means I have done the work! Could you show me how to prove it or deny it? By the way, I believe that ...
5
votes
2answers
122 views

How to prove the inequality $(1+a+a^2)(1+b+b^2)(1+c+c^2) \leq (1+a+b^2)(1+b+c^2)(1+c+a^2)?$

For $a,b,c>0$ prove the inequality $$ (1+a+a^2)(1+b+b^2)(1+c+c^2) \leq (1+a+b^2)(1+b+c^2)(1+c+a^2). $$ Seems the rearrangement inequality must help but I can't do it. Any ideas?
5
votes
3answers
86 views

How can I prove this inequality $\frac{|xy|}{2x^2+y^2}\le1 $?

How can I prove this $\frac{|xy|}{2x^2+y^2}\le1 $ ? I was thinking about considering the left term a function and maybe show that 1 is the extreme point x,y can be any real number but not 0
3
votes
1answer
51 views

I need to show that the following function is positive.

I need to show that the following function is positive. $H(x)=2(7)^x+2(4)^x-2(3)^x+2(d+1)^x+2((d-2)(d+4))^x-2(d+2)^x-2((d-1)(d+5))^x$ Where $d=3,4,5 $ and $x\in[-1,0)$ From graph for different ...
6
votes
1answer
121 views

with inequality $\frac{1}{3a+5b+7c}+\frac{1}{3b+5c+7a}+\frac{1}{3c+5a+7b}\le\frac{\sqrt{3}}{4}$

let $a,b,c>0$, such $ab+bc+ac=1$,show that $$\dfrac{1}{3a+5b+7c}+\dfrac{1}{3b+5c+7a}+\dfrac{1}{3c+5a+7b}\le\dfrac{\sqrt{3}}{4}$$ by Macavity C-S:with inequality ...
0
votes
0answers
32 views

following system of inequalities

Graph the following system of inequalities. $$x+2y\leq 12$$ $$2x+y\leq 12$$ $$x\geq 0 , y \geq 0$$ I just need to know how Can show the augmented matrix that describes the situation
0
votes
1answer
44 views

Vectorial sequence space and its inner product

For $k\in \mathbb{N}$, let $\lambda_k \in \mathbb{R}^{d\times d}$ be a symetric positive definite matrix, and $\lambda_{kij}$ be it's coordinates. Suppose we have $c_k \in \mathbb{R}^d$, $c_{ki}$ its ...
1
vote
2answers
27 views

The shaded solution region in the following graph represents the inequality

The shaded solution region in the following graph represents the inequality please help I don't know how determine that from the graph
0
votes
2answers
30 views

Graph the following system of inequalities

Graph the following system of inequalities. Show (by shading in) the feasible region. $$x+2y\leq 12$$ $$2x+y\leq 12$$ $$x\geq 0 , y \geq 0$$ I would like to know how to graph these ...
0
votes
2answers
41 views

Why does this inequality stand?

I stand that $\log n=O(n^{\epsilon})$ for any $\epsilon >0$. At a previous example we have shown that $$e^{n^{\epsilon}} \geq \frac{n^{\epsilon d}}{d!}$$ where $d=\lfloor ...
4
votes
2answers
121 views

Prove the following inequality without using induction: $\frac{1}{2^k-1}\leq \sin^{2k}\theta+\cos^{2k}\theta\leq 1$

How to prove the following inequality (without using induction)? $$\frac{1}{2^k-1}\leq \sin^{2k}\theta+\cos^{2k}\theta\leq 1,\quad k\in\Bbb N.$$