Questions tagged [cauchy-schwarz-inequality]

Problems with using C-S (Cauchy-Schwarz inequality)

Filter by
Sorted by
Tagged with
0
votes
2answers
50 views

Prove a vector integral inequality

Given continuous $f:\mathbb{R}\rightarrow\mathbb{R}^n$, $x_1<x_2$, how to prove $||\int_{x_1}^{x_2}f(x)dx|| \leq \int_{x_1}^{x_2}||f(x)||dx$? I think it needs to use Cauchy-Schwarz inequality, and ...
2
votes
3answers
173 views

Find the Maximum value of $\frac{x}{\sqrt{x+y}}+\frac{y}{\sqrt{y+z}}+\frac{z}{\sqrt{z+x}}$

if $x$, $y$ and $z$ are positive real numbers such that $x+y+z=4$ Find the maximum value of $$S=\frac{x}{\sqrt{x+y}}+\frac{y}{\sqrt{y+z}}+\frac{z}{\sqrt{z+x}}$$ I tried as follows. The given ...
19
votes
12answers
2k views

Knowing that for any set of real numbers $x,y,z$, such that $x+y+z = 1$ the inequality $x^2+y^2+z^2 \ge \frac{1}{3}$ holds.

Knowing that for any set of real numbers $x,y,z$, such that $x+y+z = 1$ the inequality $x^2+y^2+z^2 \ge \frac{1}{3}$ holds. I spent a lot of time trying to solve this and, having consulted some books,...
0
votes
4answers
80 views

Prove this inequality $\sum_{cyc}\frac{1}{x+3y}\ge\sum_{cyc} \frac{1}{z+2x+y}$

Given $x,y,z$ are positive number. Prove that $$\frac{1}{x+3y}+\frac{1}{y+3z}+\frac{1}{z+3x}\ge \frac{1}{x+2y+z}+\frac{1}{y+2z+x}+\frac{1}{z+2x+y}$$ we must prove $\frac{1}{x+2y+z}+\frac{1}{y+2z+x}+\...
1
vote
3answers
80 views

For a given condition is it true that $a+b+c=3$. [duplicate]

Suppose a, b, c are positive real numbers such that $$(1+a+b+c)\left(1+\frac 1a+\frac 1b+\frac 1c\right)=16$$ Then is it true that we must have $a+b+c=3$ ? Please help me to solve this. Thanks ...
1
vote
3answers
71 views

Find $x$ and $y$ such that $15\sin(x+y)+7\sin x+7\sin y$ is maximized

Question is: Find $x$ and $y$ such that $$15\sin(x+y)+7\sin x+7\sin y$$ is maximized. What I tried was $$f(x,y)=15\sin(x+y)+7\sin x+7\sin y=15(\sin x\cos y+\cos x\sin y)+7(\sin x +\sin y)$$ $$\frac{\...
0
votes
2answers
64 views

$ABC$ be a triangle $R_a$, $R_b$, $R_c$ are the radii of Lucas Circles of $ABC$. Prove that:

$ABC$ be a triangle $R_a$, $R_b$, $R_c$ are the radii of Lucas Circles of $ABC$. Prove that: $$R_a+R_b+R_c\geq \dfrac {8.\triangle}{(1+\sqrt {3})^2.R}$$ where $\triangle$ and $R$ are area and ...
3
votes
2answers
100 views

Find the minimum of the value $k$ such

Let $x\geq0$, $y\geq0$, $z\ge 0$ such that $$\sqrt{x^2+y^2}+\sqrt{y^2+z^2}+\sqrt{z^2+x^2}\le\sqrt{k(x^2+y^2+z^2)+(6-k)(xy+yz+xz)}$$ Find the minimum of the real value $k$ I square the side $$2(x^...
1
vote
3answers
3k views

Prove that $\frac{1}{a^3(b+c)} + \frac{1}{b^3(a+c)} + \frac{1}{c^3(a+b)} \geq \frac{3}{2} \text{ when } abc=1$

So I have a possible proof but I'm not certain it's right: $\text{As } \frac{a_1}{x_1} + \frac{a_2}{x_2} + \frac{a_3}{x_3} \geq \frac{(a_1+a_2+a_3)^2}{x_1+x_2+x_3}\text{, we obtain } \frac{1}{a^3(b+c)...
0
votes
4answers
261 views

Showing that $(x_2 - x_1)^2 + (y_2 - y_1)^2 \leq 8$, where $x^2 + y^2 \leq 1$

Let $D^2$ be the unit disk in $\mathbb{R}^2$ and $(x_1, y_1), (x_2, y_2) \in D^2$. I can't show that $$(x_2 - x_1)^2 + (y_2 - y_1)^2 \leq 8.$$ Can someone give me a hint?
1
vote
2answers
107 views

Prove this $1+2abcxyz \geq a^2x^2 + b^2y^2+c^2z^2$ [closed]

Let $a,b,c,x,y,z\in\mathbb{R}$ such that \begin{align}\{a,b,c,x,y,z\}&\subset[-1,1]\\ 1 + 2abc &\geq a^2+b^2+c^2\\ 1+2xyz&\geq x^2+y^2+z^2\end{align} Prove that: $$1+2abcxyz \geq a^2x^2 +...
0
votes
1answer
265 views

If $x + y + z + w=5$ then the minimum value of $x^2 \cot (9°) + y^2 \cot (27°) + z^2 \cot (63°) + w^2 \cot (81°)$ is?

I used the $A.M. \ge G.M.$ rule. $$((xyzw)^2 . \cot(9^o)\tan(9^o)\cot(27^o)\tan(27^o))^{1/4} \le \space \frac {x^2 \cot (9°) + y^2 \cot (27°) + z^2 \cot (63°) + w^2 \cot (81°)}{4}$$ This reduces to :...
3
votes
1answer
149 views

Maximize $P=\frac{\sqrt{a^2-1}}{a}+\frac{\sqrt{b^2-1}}{b}+\frac{\sqrt{c^2-1}}{c} $

Let $a,b,c\geq 1$ satisfy $ 32abc=18(a+b+c)+27$. Find the maximum value $$P=\dfrac{\sqrt{a^2-1}}{a}+\dfrac{\sqrt{b^2-1}}{b}+\dfrac{\sqrt{c^2-1}}{c} $$ $\sqrt {a^2-1}=\sqrt{(a-1)(a+1)}\leq \frac{a-1+a+...
4
votes
2answers
221 views

AM-GM giving wrong result on applying to trigonometric functions

Question: Find the range of $f(x)=\operatorname{cosec} ^2x+25\sec^2x$ My attempt: Applying AM-GM: $$f(x) \geq 2\sqrt{\operatorname{cosec}^2x\cdot25\cdot \sec^2x}$$ $$f(x) \geq 10\cdot \left|\...
1
vote
3answers
101 views

Maximize $P=\frac{1}{\sqrt{1+x^{2}}}+\frac{1}{\sqrt{1+y^{2}}}+\frac{1}{\sqrt{1+z^{2}}}$

For $x,y,z$ are positive real numbers that satisfy $xy+yz+xz=1$. Maximize $$P=\frac{1}{\sqrt{1+x^{2}}}+\frac{1}{\sqrt{1+y^{2}}}+\frac{1}{\sqrt{1+z^{2}}}.$$ I think if we let $x=\tan A;y=\tan B;z=\tan ...
-1
votes
1answer
69 views

Maximize $\frac{1}{\sqrt{2a^2+b^2}}+\frac{1}{\sqrt{2a^2+c^2}}+\frac{1}{\sqrt{2c^2+a^2}}$ under some condition on $(a,b,c)$

For $a,b,c$ are postive real number satisfy $7\left(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}\right)=6\left(\dfrac{1}{ab}+\dfrac{1}{bc}+\dfrac{1}{ca}\right)+2015$. Find Maximize $$P=\dfrac{1}{\sqrt{...
3
votes
3answers
237 views

If $abc=1$ so $\sum\limits_{cyc}\frac{7-6a}{2+a^2}\geq1$

Let $a$, $b$ and $c$ be real numbers such that $abc=1$. Prove that: $$\frac{7-6a}{2+a^2}+\frac{7-6b}{2+b^2}+\frac{7-6c}{2+c^2}\geq1$$ The equality occurs also for $a=b=2$ and $c=\frac{1}{4}$. This ...
1
vote
4answers
74 views

Prove the inequality, power.

$\{ x,y \in\Bbb R\ \}$ If $x+y = 2$ then prove the inequality: $x^4 + y^4 \ge 2$ How I started $(x+y)^2 = 4$ $x^2 + y^2 = 4 - 2xy$ $(x^2+y^2)^2 - 2(xy)^2 \ge 2$ $(4-2xy)^2 - 2(xy)^2 \ge 2$ $16-16xy +...
-2
votes
3answers
52 views

A Problem Involving an Inequality [closed]

How to prove that $\frac1{a^2} + \frac1{b^2} + \frac1{c^2} \geq \frac1{ab} + \frac1{bc} + \frac1{ac}$ Assume that given symbolic terms are REAL and POSITIVE
1
vote
1answer
121 views

Square root inequality cyclic

Is it true that if $a+b+c=1$, where $a$, $b$ and $c$ are nonnegative real numbers, then $$(3-2\sqrt{2})\sum\limits_{cyc}\sqrt{ab}+2\sqrt{2}-1\geq\sum\limits_{cyc}\sqrt{(1-a)(1-b)}?$$ Edit: I was told ...
1
vote
1answer
209 views

For $a,b,c$. Prove that $\frac{a^2}{a+\sqrt[3]{bc}}+\frac{b^2}{b+\sqrt[3]{ca}}+\frac{c^2}{c+\sqrt[3]{ab}}\ge\frac{3}{2}$

For $a,b,c$ are positive real number satisfy $a+b+c=3$. Prove that $$\dfrac{a^2}{a+\sqrt[3]{bc}}+\dfrac{b^2}{b+\sqrt[3]{ca}}+\dfrac{c^2}{c+\sqrt[3]{ab}}\ge\dfrac{3}{2}$$ By Cauchy-Schwarz: $\...
0
votes
1answer
83 views

For $a,b,c>0$ prove that $\frac{a}{a+\sqrt{(a+b)(a+c)}} + \frac{b}{b+\sqrt{(b+a)(b+c)}}+ \frac{c}{c+\sqrt{(c+b)(c+a)}} \leq 1$ [closed]

For $a,b,c>0$ prove that $$\frac{a}{a+\sqrt{(a+b)(a+c)}} + \frac{b}{b+\sqrt{(b+a)(b+c)}} +\frac{c}{c+\sqrt{(c+b)(c+a)}} \leq 1$$ Taken from local contest. I have no clue on how to approach this ...
3
votes
6answers
153 views

Proving: $\left(a+\frac1a\right)^2+\left(b+\frac1b\right)^2\ge\frac{25}2$ [duplicate]

For $a>0,b>0$, $a+b=1$ prove:$$\bigg(a+\dfrac{1}{a}\bigg)^2+\bigg(b+\dfrac{1}{b}\bigg)^2\ge\dfrac{25}{2}$$ My try don't do much, tough $a+b=1\implies\dfrac{1}{a}+\dfrac{1}{b}=\dfrac{1}{ab}$ ...
1
vote
4answers
89 views

Hint needed to show that $a\cos t+b\sin t\leq \sqrt{a^2+b^2}$ for $t\in [0,2\pi)$ [duplicate]

And that the upper bound is achieved for some choice of $\theta$. This exercise shows up in the Cauchy-Schwarz section of a textbook I am looking through but I don't see how to apply CS to prove. I ...
0
votes
4answers
94 views

Inequalities help: $(a^3+b^3)^2\leq (a^2+b^2)(a^4+b^4)$

Prove: that $$(a^3+b^3)^2\leq (a^2+b^2)(a^4+b^4)$$ for all real numbers $a$ and $b$. My attempt: $(a^2+b^2)(a^4+b^4)=a^6+a^2 b^4+b^2 a^4+b^6\geq 4\cdot(a^6\cdot b^6\cdot a^2 b^4\cdot a^4 b^2)^{\frac{...
3
votes
8answers
113 views

Finding maximum of $x+y$ [closed]

Let x and y be real numbers satisfying $9x^{2} + 16y^{2} = 1$. Then $x + y$ is maximum when a. $y = \frac{9x}{16}$ b. $y = −\frac{9x}{16}$ c. $y = \frac{4x}{3}$ d. $y = −\frac{4x}{3}$
2
votes
4answers
180 views

Let $a,b,c\in \Bbb R^+$ such that $(1+a+b+c)(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})=16$. Find $(a+b+c)$

Let $a,b,c\in \Bbb R^+$ such that $(1+a+b+c)(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})=16$. Find $(a+b+c)$. I computed the whole product ;If $(a+b+c)=x\implies (1+x)(1+\frac{bc+ca+ab}{abc})=16$. ...
3
votes
3answers
93 views

Prove that $n \geq \sqrt{n-1} + \sqrt{n}$ for $n \geq 4$

I am trying to prove that $n \geq \sqrt{n+1} + \sqrt{n}$ for $n \geq 4$ (n in naturals of course). I am not sure if there are any specific inequalities that could help me out here. I also know that 4 ...
4
votes
2answers
60 views

Triangular inequality extended

Let $a,b,c,d,e,f\in\mathbb{R}$ all positive or zero, such that $a\leq c+d$ and $b\leq e+f$ show that: $$\sqrt{a^2+b^2}\leq\sqrt{c^2+e^2}+\sqrt{d^2+f^2}$$ Some hint or auxiliar inequality that would ...
1
vote
3answers
84 views

Minimize $P=5\left(x^2+y^2\right)+2z^2$

For $\left(x+y\right)\left(x+z\right)\left(y+z\right)=144$, minimize $$P=5\left(x^2+y^2\right)+2z^2$$ I have no idea. Can you make a few suggestions?
1
vote
2answers
112 views

Prove the inequality, fractions.

$\{ a,b,c \in\Bbb R_+\ \}$ If $\frac {1}{ab} +\frac {1}{bc} + \frac {1}{ac} = 3$ then prove the inequality: $ab + bc + ac \ge 3$ How I started $ab + bc + ac \ge \frac {1}{ab} +\frac {1}{bc} + \frac {...
2
votes
3answers
127 views

Prove that $\frac{a}{\sqrt{a^2+b^2}}+\frac{b}{\sqrt{9a^2+b^2}}+\frac{2ab}{\sqrt{a^2+b^2}\times \sqrt{9a^2+b^2} }\leq \frac32.$

Prove that $$\dfrac{a}{\sqrt{a^2+b^2}}+\dfrac{b}{\sqrt{9a^2+b^2}}+\dfrac{2ab}{\sqrt{a^2+b^2}\times \sqrt{9a^2+b^2} }\leq \dfrac{3}{2}.$$ When is equality attained ? My Attempt : I could not ...
6
votes
1answer
173 views

An inequality concerning Lagrange's identity

Does the following inequality still hold $$(a^2_{1}+b^2_{2}+b^2_{3})(a^2_{2}+b^2_{3}+b^2_{1})(a^2_{3}+b^2_{1}+b^2_{2})\ge (b^2_{1}+b^2_{2}+b^2_{3})(a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3})^2 $$ $$+\dfrac{...
3
votes
1answer
85 views

If $x,y,z>0$, prove that $\frac{x+y+z}{3\sqrt{3}} \geq \frac{yz+zx+xy}{\sum \limits_{cyc} \sqrt{x^2+xy+y^2}}$

If $x,y,z > 0$, prove that $\dfrac{x+y+z}{3\sqrt{3}} \geq \dfrac{yz+zx+xy}{ \sqrt{x^2+xy+y^2}+\sqrt{y^2+yz+z^2}+\sqrt{z^2+zx+x^2}}$ with equality if and only if $x=y=z$. SOURCE :CRUX (...
2
votes
6answers
101 views

Minimize $\big(3+2a^2\big)\big(3+2b^2\big)\big(3+2c^2\big)$ if $a+b+c=3$ and $(a,b,c) > 0$

Minimize $\big(3+2a^2\big)\big(3+2b^2\big)\big(3+2c^2\big)$ if $a+b+c=3$ and $(a,b,c) > 0$. I expanded the brackets and applied AM-GM on all of the eight terms to get : $$\big(3+2a^2\big)\big(3+...
1
vote
1answer
93 views

Proving an inequality via the Cauchy-Schwarz Inequality

I apolgize for contributing yet another question asking about an application of CS. Here it is: Suppose $p_1, \dots ,p_n$ and $a_1,...,a_n$ are real numbers such that $p_i \geq 0$, $a_i \geq 0$ for ...
3
votes
2answers
1k views

Finding maxima and minima of $f(x,y)= x^4 + y^4 - 2x^2 - 2y^2 + 4xy$

For $f(x,y)=x^4+y^4-2x^2-2y^2+4xy$, I need to find maxima or minima. There are three critical points: $(0,0),(\sqrt2, -\sqrt2),(-\sqrt2,\sqrt2)$ So at $(\sqrt2, -\sqrt2)$, $f$ has minimum value, $-8$ ...
6
votes
2answers
205 views

How to show this inequality $\sum\sqrt{\frac{x}{x+2y+z}}\le 2$

Let $x,y,z,w>0$ show that $$\sqrt{\dfrac{x}{x+2y+z}}+\sqrt{\dfrac{y}{y+2z+w}}+\sqrt{\dfrac{z}{z+2w+x}}+\sqrt{\dfrac{w}{w+2x+y}}\le 2$$ I tried C-S, but without success.
3
votes
3answers
80 views

maximum value of expression $6bc+6abc+2ab+3ac$

If $a,b,c>0$ and $a+2b+3c=15,$ then finding maximum value of $6bc+6abc+2ab+3ac$ is with the help of AM - GM inequality $4ab\leq (a+b)^2$ and $4bc \leq (b+c)^2$ and $\displaystyle 4ca \leq (c+a)^...
0
votes
7answers
107 views

Difficult trigonometry problem to find the minimum value

Find the minimum value of $5\cos A + 12\sin A + 12$. I don't know how to approach this problem. I need help. I'll show you how much I got... $$5\cos A +12\sin A + 12 = 13(5/13\cos A +12/13\sin A) + ...
3
votes
1answer
187 views

Prove $\frac{ab}{1+a+b}+\frac{bc}{1+b+c}+\frac{ac}{1+a+c}\geq \frac{3}{2}$ for $a$, $b$, $c$ positive and $1+a+b+c=2abc$

Given $1+a+b+c = 2abc$ and positivity of real numbers $a,b,c$, we are asked to prove that $$\frac{ab}{1+a+b}+\frac{bc}{1+b+c}+\frac{ac}{1+a+c}\geq \frac{3}{2}$$ If $d=a+b+c$ I got as far as to ...
5
votes
5answers
251 views

If $a+b+c = 6$ and $a$, $b$, $c$ are nonnegative then $a^2+b^2+c^2 \geq 12$ [duplicate]

Let $a,b,c$ be three positive real numbers such that $a+b+c = 6$. Prove that $a^2+b^2+c^2 \geq 12$. I tried using the AM-GM inequality to solve the same, however I wasn't able to make any ...
2
votes
1answer
65 views

How prove $\sum\limits_{cyc}\sqrt{a^2+b^2-c^2}\sqrt{a^2-b^2+c^2}\le ab+bc+ca$

Let acute-angled triangle $ABC$,and $AB=c,BC=a,AC=b$,show that $$\sum_{cyc}\sqrt{a^2+b^2-c^2}\sqrt{a^2-b^2+c^2}\le ab+bc+ca$$ I try use AM-GM $$\sum_{cyc}\sqrt{a^2+b^2-c^2}\sqrt{a^2-b^2+c^2}\le\sum_{...
4
votes
2answers
169 views

AM-GM inequality: $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{d} + \frac{d^2}{a} \geq a + b + c + d$

Let $a, b, c, d > 0$. Prove that $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{d} + \frac{d^2}{a} \geq a + b + c + d$. I'm supposed to prove this by AM-GM, but I can't see how. Any help would be ...
2
votes
3answers
92 views

Proving $\sqrt{x+y-x^2}+\sqrt{y+z-y^2}+\sqrt{z+w-z^2}+\sqrt{w+x-w^2}\le4\sqrt2-3$

I found this inequality using unusual calculations in maths Olympics and I wonder if some clever teenager could prove it using their elementary knowledge of mathematics. Let $x,y,z,w$ be non-...
4
votes
1answer
274 views

Given $ab+bc+ca=3abc$, prove $\sqrt{\frac{a+b}{c(a^2+b^2 )}}+\sqrt{\frac{b+c}{a(b^2+c^2)}}+\sqrt{\frac{c+a}{b(c^2+a^2 )}}\leq 3$

$a, b, c$ are positive real numbers such that $ab+bc+ca=3abc$ Prove∶ $$\sqrt{\frac{a+b}{c(a^2+b^2 )}}+\sqrt{\frac{b+c}{a(b^2+c^2)}}+\sqrt{\frac{c+a}{b(c^2+a^2 )}}\;\;\leq\; 3$$
-2
votes
2answers
115 views

If $xy+xz+yz=3$ so $\sum\limits_{cyc}\left(x^2y+x^2z+2\sqrt{xyz(x^3+3x)}\right)\geq2xyz\sum\limits_{cyc}(x^2+2)$

Prove that for any set of three positive real $x, y, z$ such that $xy+yz+zx=3$ $x^2(y+z)+y^2(x+z)+z^2(x+y)+2\sqrt {xyz}\left(\sqrt{x^3+3x}+\sqrt{y^3+3x}+\sqrt{z^3+3x}\right)\ge$ $\ge 2xyz(x^2+y^2+z^2+...
3
votes
1answer
123 views

Prove the inequality $\sum\limits_{cyc}\sqrt{x+yz}\ge\sqrt{xyz}+\sqrt{x}+\sqrt{y}+\sqrt{z}$

Let $x,y,z$ is positive real numbers, such that $\frac1x+\frac1y+\frac1z=1$. Prove the inequality $$\sqrt{x+yz}+\sqrt{y+zx}+\sqrt{z+xy}\ge\sqrt{xyz}+\sqrt{x}+\sqrt{y}+\sqrt{z}$$ My work so far: $\...
4
votes
1answer
141 views

How prove this converse cauchy inequality of problem?

Problem: let $A$ and $B$ be positive real numbers such that $$-A\le a_{i}\le A,-B\le b_{i}\le B~(i=1,2,\cdots,n)$$. Show that $$n\sum_{i=1}^{n}a^2_{i}b^2_{i}-\left(\sum_{i=1}^{n}a_{i}b_{i}\...
4
votes
0answers
2k views

If $a, b, c$ are sides of a triangle, prove that $\frac{a}{b+c-a} + \frac{b}{a+c-b} + \frac{c}{a+b-c} \ge 3$ [duplicate]

If $a, b, c$ are sides of a triangle, how can I prove that $$\frac{a}{b+c-a} + \frac{b}{a+c-b} + \frac{c}{a+b-c} \ge 3$$