Questions tagged [a.m.-g.m.-inequality]

For questions about proving and manipulating the AM-GM inequality. To be used necessarily with the [inequality] tag.

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How can i find minimum value of this inequality

$a_1,a_2,...,a_n$ are 8 distinct positive integers. $b_1,b_2,...,b_n$ are another 8 distinct positive integers ($a_i,b_j$ are not necessarily y distinct for $i, j = 1, 2, ...8$).Enter the smallest ...
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Let m be the smallest number among: $(x-y)^2, (y-z)^2, (z-x)^2$ Prove $m \le \frac{1}{2}(x^2+y^2+z^2)$

Let m be the smallest number among: $(x-y)^2, (y-z)^2, (z-x)^2$ Prove $m \le \frac{1}{2}(x^2+y^2+z^2)$ My attempts: $3m \le (x-y)^2+(y-z)^2+(z-x)^2$ so I tried to prove $(x-y)^2+(y-z)^2+(z-x)^2 \le \...
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3 votes
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A conditional negative definite quadratic form involving $\ln$ function

This is a follow up of a recent question that has been closed because the claimed property has counter-examples. The question was asked under this form: Given $a_1,\ldots,a_n,x_1,\ldots,x_n\in\mathbf{...
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Find bounds of $\prod_{i=1}^k (x_i-x_{i+1})$ where $\sum_{i=1}^k x^2_i=1$

Let $x_1,x_2,...,x_k$ be real numbers such that $\sum_{i=1}^k x^2_i=1$. Determine the minimum and maximum (if there is) value of $$\prod_{i=1}^k (x_i-x_{i+1})$$ and determine all values of $(x_1,x_2,...
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Prove that for positive reals $x,y,z$, $x^6+y^6+z^6 + 6x^2y^2z^2 \geq 3xyz(x^3+y^3+z^3)$.

I am not sure if the inequality is true. My first attempt was to try AM-GM inequality in clever ways. I also tried Schur's inequality which gives $$ x^6+y^6+z^6 + 6x^2y^2z^2 \geq (x^2+y^2+z^2)(x^2y^2+...
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Prove that $(1+x)^k/k + (1-x)^m/m\geq 1/k +1/m$ without calculus

Note: this has been edited to make the question more general. I want to show that $(1+x)^k/k + (1-x)^m/m$ is minimized at $x=0$ when $k,m\geq 1$ and $-1\leq x \leq1$. Of course, I could take the first ...
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2 votes
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How can I prove that the perimeter is at most 60?

Problem: Let $\Delta$ be a triangle in the plane. Let $P$ be the perimeter of the triangle and $A$ be the area. Let $a,b,c$ be the length of the sides and suppose they are positive integers. Suppose ...
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Inequality of positive integers [duplicate]

I have to prove the following inequalities for positive integer $n$: $$n^n\geqslant\left(\frac{n+1}2\right)^{n+1}$$ $$ 2^ {n(n+1)}> (n+1)^{n+1} \left(\frac{n}{1}\right)^{n} \left(\frac{n-1}{2}\...
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3 answers
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find the maximum of $x+\frac{1}{2y}$ under the condition $(2xy-1)^2=(5y+2)(y-2)$

Assume $x,y \in \mathbb{R}^+$ and satisfy $$\left(2xy-1\right)^2 = (5y+2)(y-2)$$, find the maximum of the expression $x+\frac{1}{2y}$. $\because (5y+2)(y-2) \geq 0$ , $\therefore y\geq 2$ or $y \leq -\...
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To prove the inequality of positive rational numbers

Show that: $$ \left(\frac{a+b}{a+b+c}\right)^{c} \left(\frac{b+c}{a+b+c}\right)^{a} \left(\frac{a+c}{a+b+c}\right)^{b}< \left(\frac{2}{3}\right)^{a+b+c} ,a\ne b\ne c$$ PS: I am supposed to use ...
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2 votes
1 answer
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Is there a solution for $(\frac{3}{x+y+z})^n+(\frac{3}{x+y+z})^{5-n}<2$?

Is there a solution for $$\left(\frac{3}{x+y+z}\right)^n+\left(\frac{3}{x+y+z}\right)^{5-n}<2$$ where $n\in\mathbb Z$ and $x,y,z>0, xyz=1$. This is the part of my attempts for my homework, that ...
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How would I prove the following using the AGM inequality?

Question 17. Let $x,y\in\mathbb R$, $x,y\geq0$. Prove that $$(\sqrt x+\sqrt y)^2\geq2\sqrt{2(x+y)\sqrt{xy}}.$$ I believe I have to use AGM multiple times, but I am not exactly sure how
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2 votes
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Proving $\frac{x^2+y^2+z^2}{x+y+z}+\frac 32\sqrt[3]\frac{xy+yz+xz}{x+y+z}\geq \frac 52$, for positive values with $xyz=1$, without expansion?

Let, $x,y,z>0$ and $xyz=1$, then prove that $$\frac{x^2+y^2+z^2}{x+y+z}+\frac 32\sqrt[3]\frac{xy+yz+xz}{x+y+z}\geq \frac 52$$ I know that $$x^2+y^2+z^2\geq x+y+z$$ by Cauchy-Schwarz. So, $$\frac{x^...
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  • 450
5 votes
2 answers
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How can I prove that, if $x,y,z>0$ and $xyz=1$, then $2(x^2+y^2+z^2)+9\geq 5(x+y+z)$

How can I prove that, if $x,y,z>0$ and $xyz=1$, then $$2(x^2+y^2+z^2)+9\geq 5(x+y+z)$$ I used the famous inequality $$x^2+y^2+z^2+3\geq 2(x+y+z)$$ I got $$2(x^2+y^2+z^2)+9\geq 4(x+y+z)+3\geq 5(x+y+...
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  • 450
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To prove the relation between n numbers and their AM - GM [duplicate]

Given $A$ and $G$ to be the arithmetic and geometric mean of n positive real numbers $a_1, a_2,...,a_n$ then for any $k > 0$ show that $$ (k+A)^n \ge\ (k+a_1)...(k+a_n) \ge\ (k+G)^n .$$ I started ...
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  • 375
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2 answers
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To prove the following inequalities of positive rational numbers

I have to prove the following inequalities: $$ a^ab^bc^c \ge \ (\frac{a+b}{2})^{\frac{a+b}{2}} (\frac{c+b}{2})^{\frac{c+b}{2}} (\frac{a+c}{2})^{\frac{a+c}{2}} $$ $$(a+b)^{c}(c+b)^{a}(a+c)^{b} < \...
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  • 375
0 votes
1 answer
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To prove the following inequality of real numbers

If $a_1,...a_n$ are all positive real numbers then prove that $$\left(\frac{a_1 + a_2 + \dots + a_n}{n}\right)^n \ge a_1a_2\left(\frac{a_3 + a_4 + \dots + a_n}{n-2}\right)^{n-2}.$$ I approached the ...
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  • 375
3 votes
2 answers
103 views

To prove the given inequality of n numbers

Given $ a_1 + a_2 + ... + a_n = S$ where $a_1,...,a_n$ are positive reals and all $a_i$s are not equal, then show that $$ \prod_{i=1}^n\dfrac{S-a_i}{n-1} > a_1.a_2...a_n$$ I began by considering ...
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  • 375
0 votes
1 answer
25 views

Prove the A.M.-G.M. inequality for n terms using induction

You are required to give a detailed proof of the A.M.-G.M. inequality using induction. I have one answer and I am posting it but it is too brutish. I want a more elegant method of proving A.M.-G.M. ...
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Proving the AM-GM Inequality by induction [closed]

I am required to prove the AM-GM inequality using induction but via this route: (i) Let $a_1$, $a_2$, ..., an be a sequence of positive numbers. Denote their sum by $s$ and their geometric mean by $G$....
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question based on AM/GM [closed]

If $a, b, c$ are positive real numbers and \begin{align*} a^{2}(1 + b^{2})+ b^{2}(1 + c^{2}) + c^{2}(1 + a^{2}) = 6abc \end{align*} Find $a + b + c$.
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1 answer
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Solve the inequality $ \sum_{cyc}\frac{a-bc}{a+bc} \le \frac32$

Solve the inequality $ \displaystyle\sum_{cyc}\frac{a-bc}{a+bc} \le \frac32$ given $a + b + c = 1$ and $a, b, c \in \mathbb{R_{>0}}$ So, I wanted to use the known inequality $9(a+b)(b+c)(c+a) \ge ...
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4 votes
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Prove the inequality $9(a+b)(b+c)(c+a) \geq 8(a+b+c)(ab+bc+ca)$

Prove the inequality $9(a+b)(b+c)(c+a) \geq 8(a+b+c)(ab+bc+ca)$ for $a, b, c \in \mathbb{R_{>0}}$ I tried by first using AM-HM inequality on $a, b, c$ to get the following result. $\frac{a+b+c}3 \...
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  • 86
1 vote
1 answer
73 views

Prove $\sum_{j=1}^{n^2} \log_n(2j-1)\leq2n^2 $

Deduce whether the statement is true or false. Suppose $n\in \mathbb N \setminus \{0,1\}$. Then, $$\sum_{j=1}^{n^2} \log_n(2j-1)\leq2n^2 $$ I would like to ask what inequality I can apply or any ...
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  • 149
3 votes
2 answers
127 views

Showing $\frac{xy+yz+xz}{x+y+z}>5-\sqrt{4(x^2+y^2+z^2)+6}$ for $x,y,z>0$ and $xyz=1$

Let $x,y,z>0$ with $ xyz=1$ then prove that, $$\frac{xy+yz+xz}{x+y+z}>5-\sqrt{4(x^2+y^2+z^2)+6}$$ Let $$5≤\sqrt{4(x^2+y^2+z^2)+6}\implies x^2+y^2+z^2≥\frac {25-6}{4}=\frac {19}{4}$$ then the ...
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  • 450
2 votes
3 answers
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Prove that $(xy+yz+xz)(x^2+y^2+z^2+x+y+z)≥2(x+y+z)^2$

Let, $x,y,z>0$ such that $xyz=1$, then prove that $$(xy+yz+xz)(x^2+y^2+z^2+x+y+z)≥2(x+y+z)^2$$ My progress: Using the Cauchy-Schwars inequality I got $$(xy+yz+xz)(x^2+y^2+z^2+x+y+z)≥2(xy+yz+xz)(x+...
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  • 450
1 vote
2 answers
109 views

Is this nice-looking inequality actually trivial?

Let, $x,y,z>0$ such that $ xyz=1$, then prove that $$(xy+yz+xz)(x^2+y^2+z^2+xy+yz+xz)≥2(x+y+z)^2 $$ I tried to use the inequality $$x^2+y^2+z^2≥xy+yz+xz$$ Then I got, $$(xy+yz+xz)(x^2+y^2+z^2+xy+...
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  • 450
2 votes
2 answers
56 views

Can we prove the inequality without opening the parentheses? $(x+y+z)(xy+yz+xz)(x^2+y^2+z^2)≥6(x^2+y^2+z^2)+3(xy+yz+xz)$

Let, $x,y,z>0$ such that $ xyz=1$, then prove that $$(x+y+z)(xy+yz+xz)(x^2+y^2+z^2)≥6(x^2+y^2+z^2)+3(xy+yz+xz)$$ I tried to use the following inequalities: $$x^2+y^2+z^2≥xy+yz+xz$$ and The Cauchy–...
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  • 450
3 votes
2 answers
60 views

Find minimum of the function using AM-GM

Problem: Find the minimum of the function $f(x,y)=x + \frac{8}{y(x-y)}$, where $x>y>0$ using AM-GM. My attempt: $$f(x,y)=2\cdot \frac{x+\frac{8}{y(x-y)}}{2} \ge 2 \sqrt{\frac{8x}{y(x-y)}}$$ But ...
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0 votes
1 answer
32 views

Is the geometric mean bounded above by this value?

It is clear that the geometric mean is bounded above by the arithmetic mean: $$ \prod_{k=1}^{M} x_k^{\alpha_k} \leq \sum_{k=1}^{M}\alpha_k x_k $$ Moreover, it is clear that the arithmetic mean is ...
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  • 507
-1 votes
2 answers
50 views

Alternate proofs of the inequality $\sum_{\text{cyclic}} \frac{x}{2x+y+z}\le \frac{3}{4}$.

If $x,y,z\in(0,\infty),$ prove the inequality, $$\sum_{\text{cyclic}} \frac{x}{2x+y+z}\le \frac{3}{4}$$ I have a solution using the substitution $x+y=a,$ $y+z=b$ and $z+x=c$. $$\frac{x}{2x+y+z}+\frac{...
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1 vote
1 answer
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Prove $\left(\frac{a+b+c}{3}\right)^p\leq \frac{a^p+b^p+c^p}{3}$.

Prove: Let $p$ be an integer greater than $1$. Suppose $a,b,c$ be positive real numbers. Then $\left(\frac{a+b+c}{3}\right)^p\leq \frac{a^p+b^p+c^p}{3}$. By AM-GM, I get $\frac{a+b+c}{3}\geq (abc)^{1/...
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  • 149
1 vote
1 answer
89 views

Prove that for $a,b,c \in \mathbb{R}^+$ with $abc = 1$, $\frac{1}{a^3(b + c)} + \frac{1}{b^3(a + c)} + \frac{1}{c^3(a + b)} \ge \frac{3}{2}$ [duplicate]

I would like confirmation that I did this proof correctly. If I did, it would be a milestone in my mathematical journey as it would be my first IMO problem. By Cauchy, we have that $$\left( \frac{1}{a^...
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1 vote
3 answers
110 views

What is the minimum value of $\frac{2a}{a+2b+2c} + \frac{2b}{b+2c+2a} +\frac{2c}{c+2a+2b}$ given $ a+b+c=2$?

What is the minimum value of $\frac{2a}{a+2b+2c} + \frac{2b}{b+2c+2a} +\frac{2c}{c+2a+2b}$ given $ a+b+c=2$? I know the answer is going to be $\frac{6}{5}$ because it will occur when there is equality ...
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1 vote
0 answers
68 views

Prove that for all $a,b,c \in \mathbb{R}^+$, $\sqrt[3]{\frac ab} + \sqrt[5]{\frac bc} + \sqrt[7]{\frac ca} > \frac 52$

Prove that for all $a,b,c \in \mathbb{R}^+$, $\sqrt[3]{\dfrac ab} + \sqrt[5]{\dfrac bc} + \sqrt[7]{\dfrac ca} > \dfrac 52$. My thought process along with the proof: Since we're dealing with ...
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-4 votes
3 answers
53 views

Proof based on inequalities [closed]

I want to find the minimum value of $$\frac{(5+x)(2+x)}{1+x}.$$ I brought it down to the fact that it depends on the value of $$(x^2 + x)\sqrt{7x+10}.$$ Also $x\geq -10/7$, but don't know what to do ...
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1 vote
0 answers
66 views

Maximizing $\dfrac{4 \sqrt{a} + 6 \sqrt{b} + 12 \sqrt{c}}{\sqrt{abc}}$ over $a+b+c=4abc$ and $a,b,c>0$

Let $a,$ $b,$ $c$ be positive real numbers such that $a + b + c = 4abc.$ Find the maximum value of $$\dfrac{4 \sqrt{a} + 6 \sqrt{b} + 12 \sqrt{c}}{\sqrt{abc}}.$$ First, I tried using $\dfrac{4 \sqrt{...
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1 vote
1 answer
37 views

Maximize $\lambda$ over $a^2 + b^2 + c^2 + d^2 \ge ab + \lambda bc + cd$ and $a,b,c,d\geq 0$.

Find the largest real number $\lambda$ such that $$a^2 + b^2 + c^2 + d^2 \ge ab + \lambda bc + cd$$ for all nonnegative real numbers $a,$ $b,$ $c,$ $d.$ I tried using AM-GM on like $a^2+\dfrac{b^2}{4}...
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1 vote
1 answer
57 views

Hard AM-GM Inequality

Find the sum of all positive integers $n,$ where the inequality $\sqrt{a + \sqrt{b + \sqrt{c}}} \ge \sqrt[n]{abc}$ holds for all nonnegative real numbers $a,$ $b,$ and $c.$ I tried squaring both sides ...
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-2 votes
1 answer
39 views

question about AM-GM inequality ab<(a^2+b^2)/2 [closed]

I have a question about the AM-GM inequality if a,b>0 then ab≤(a+b)^2/4; ab≤a^2+b^2; I wonder if these 2 inequality are true for all a,b>0, if not which one is correct?
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0 votes
0 answers
67 views

Proof that $\frac{x}{\sqrt y}+\frac{y}{\sqrt z}+\frac{z}{\sqrt x} \geq 3$ for $x+y+z=3$ [duplicate]

If $x+y+z=3$, where $x$, $y$ and $z$ are sides of a triangle, prove that $$\frac{x}{\sqrt y}+\frac{y}{\sqrt z}+\frac{z}{\sqrt x} \geq 3$$ We have $0< x,y,z<\frac{3}{2}$ from the triangle ...
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  • 423
0 votes
1 answer
84 views

Show that $x^x(1-x)^{(1-x)} \geq \frac{1}{2}$ for $x \in (0,1)$.

My approach: First assume $x$ is a rational. Let $x=\frac{a_1}{b_1}$ and $1-x = \frac{a_2}{b_2}$, where $a_1,a_2,b_1,b_2 \in \mathbb{Z}^{+}, a_1 < b_1, a_2<b_2.$ Somehow show that $$\left(\frac{...
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0 votes
1 answer
81 views

A friends proof of AM-GM inequality

We are supposed to prove that $ \frac{\sum_{k=1}^{n}x_k}{n}\geq (\prod_{k=1}^{n} x_k)^{1/n}$, when $x_i\geq 0$ for all $n\in\mathbb{N}$. My friend proved it for $n=2$, $n=3$ and then for an arbitrary ...
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1 vote
0 answers
34 views

inverse of AM-GM inequality for converting sums to products [closed]

I have an inequality in the form $\alpha + \sqrt{\beta}$ where both $\alpha,\beta \in [0,1]$ I want to find an upper bound for this quantity in terms of the product. Is there any inequality in the ...
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  • 1,404
2 votes
1 answer
122 views

Find the maximum value of $x^2y^3$ subject to the condition that $3x+2y=1$

I am trying to use G.M.$\leq $ A.M.as follows $(\frac{2x^2y^3}{4})^\frac{1}{5} \leq \frac{x+2x+y+\frac{y}{2} +\frac{y}{2}}{5}$ $\implies (\frac{x^2y^3}{2})^\frac{1}{5} \leq \frac{1}{5}$ $\implies$ $x^...
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  • 73
0 votes
1 answer
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What is the fastest elementary proof of $A_n≥G_n$?

The $A_m≥G_m$ has a whole lot of proofs, the large majority I've seen by induction. Currently the quickest one that I've seen is the following: (note $A_n=\frac{a_1+a_2+a_3+...+a_n}{n}, G_n=\sqrt[n]{...
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  • 1,073
1 vote
2 answers
108 views

$a + b + c + d = 1$ and $a^2 + b^2 + c^2 + d^2 = \frac{1}{3}$, where $-1 \le a,b,c,d \le 1$. Which value of $a$ is the largest possible?

I'm a complete novice who honestly has no clue about how to solve these types of problems. This question was originally multiple choice, and I was able to get the answer $\frac12$ simply through ...
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2 votes
0 answers
91 views

How to prove $\dfrac{a+b}{2}\geq\sqrt{ab}$ using ellipse

Noted that ellipse properties $d_1+d_2=2D$, focal length, $f=c$ and radius of minor axis $=r$. Let $d_1=a;d_2=b$ If $ab$ is not maximum, then $\sqrt{ab}$ not maximum W.l.o.g, prove max $ab=D^2$ $ab=\...
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  • 115
1 vote
4 answers
76 views

Find $a$ such that ${x_1}^2+{x_2}^2$ takes the minimal value where $x_1, x_2$ are solutions to $x^2-ax+(a-1)=0$ DO NOT USE CALCULUS

My thinking: Let $x_1 = \frac{a+\sqrt{a^2-4a+4}}{2}$ and $x_2 = \frac{a-\sqrt{a^2-4a+4}}{2}$ By the AGM (Arithmetic-Geometric Mean Inequality): We have $x_1\cdot x_2\le \left(\frac{x_1\cdot \:x_2}{2}\...
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  • 387
1 vote
2 answers
116 views

Let $a,b\in \mathbb{R}$ such that $ab=2$. Find the max value of $\frac{3}{2\left(a+b\right)^2}$ and $a, b$ where max is attained, without calculus.

My thinking: By Arithmetic and Geometric mean inequality (AGM): $ab\le \left(\frac{a+b}{2}\right)^2$ We know $ab=2$ $\rightarrow$ $2\le \left(\frac{a+b}{2}\right)^2$ $\rightarrow$ $2\le \frac{a^2+b^2}...
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