Basic Inequality

Algebra Level 2

$\dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{d} + \dfrac{d}{a}$

If $a, b, c$ and $d$ are any four positive real numbers, then find the minimum value of the expression above.

The answer is 4.

6 solutions

Abdur Rehman Zahid
Dec 13, 2014

Let $\color{#3D99F6}{x_0=\frac{a}{b},x_1=\frac{b}{c},x_2=\frac{c}{d},x_3=\frac{d}{a}}$ .Then by application of AM-GM we have: $\color{#D61F06}{\frac{x_0+x_1+x_2+x_3}{4}\geq\sqrt[4]{x_0x_1x_2x_3}\\x_0+x_1+x_2+x_3\geq4\sqrt[4]{\frac{a}{b}\times\frac{b}{c}\times\frac{c}{d}\times\frac{d}{a}}\\\boxed{x_0+x_1+x_2+x_3\geq4}}$

Kartik Sharma
Dec 6, 2014

This is simple, direct application of Re-arrangement inequality.

$\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} \geq \frac{a}{a} + \frac{b}{b} + \frac{c}{c} + \frac{d}{d}$

$\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} \geq 4$

What is Re-arrangement inequality?

Akshat Jain - 5 years, 6 months ago

It is a similar inequality to the am gm if you know that.

Mardokay Mosazghi - 5 years, 6 months ago

Ayush G Rai
Apr 23, 2016

guessed every variable a,b,c,d to be 1. so when you substitute we get the answer as 4

Also the fact that $\\a$ , $b$ , $c$ , $d$ all have to equal in order to achieve the minimum value.

Bloons Qoth - 4 years, 5 months ago

Haha thats funny (+1)

Ashish Menon - 5 years ago

my first solution!!!!!!!!!!!

Ayush G Rai - 5 years ago

Oh I see. Congo

Ashish Menon - 5 years ago

@Ashish Menon – thanks!!!!

Ayush G Rai - 5 years ago

can tell how to make a diagram using geogebra???

Ayush G Rai - 5 years ago

Tobias Barblan
Aug 8, 2018

By taking the gradient of this expression, equalling it to 0 and doing some rearranging, one finds that: $a=b=c=d$ and with that the expression reduces to 4.

Emmanuel Torres
Feb 8, 2017

All the variables had the same relationship to each other, so I didn't see why they should be different. 1+1+1+1=4. Sometimes simple is the most efficient.

Their average value is 1.So the expression above's denominator is 4 and 4*1=4

John Oladokun - 1 year, 2 months ago

Nihar Mahajan
Dec 5, 2014

by AM - GM inequality of size 4 to following -

( a^2dc +b^2ad + c^2ab + d^2bc ) / 4 >= ( a^4 * b^4 * c ^4 * d^4 ) ^ (1/4)

this gives us -

( a^2dc +b^2ad + c^2ab + d^2bc ) >= 4 * (abcd)

( a^2dc +b^2ad + c^2ab + d^2bc ) / abcd >= 4

(a^2dc) / abcd +(b^2ad) / abcd + (c^2ab) / abcd + (d^2bc) / abcd >= 4

hence , after reducing the fractions we get , a/b + b/c + c/d + d/a >= 4

This is a relic of the time you didn't know Latex.

Abdur Rehman Zahid - 5 years, 3 months ago

$\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{ YES!!! \ but \ now \ I \ know \ it \ well!}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$

Nihar Mahajan - 5 years, 3 months ago

$\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{ \Huge{\ddot\omega}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$

Abdur Rehman Zahid - 5 years, 3 months ago

@Abdur Rehman Zahid – $\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{ \Huge{..ping}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$

Nihar Mahajan - 5 years, 3 months ago

@Nihar Mahajan – $\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{\boxed{ \Huge{Pong}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$

Abdur Rehman Zahid - 5 years, 3 months ago

Try using latex bro.

Ayush G Rai - 5 years ago

He didn't know Latex at that time.

Abdur Rehman Zahid - 5 years ago

Basic Inequality

The answer is 4.

6 solutions

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