Question

If x, y, z are real positive numbers then minimum value of the expression

$\frac{x + y}{z} + \frac{y + z}{x} + \frac{z + x}{y} \: is \: - - - - -$
EVALUATE

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Answer 1

Answer:

x + y/z + y + z/x + z + x/y

Multiply throughout by xyz

xyz × x + y/z + xyz × y + z/x + xyz × z + x/y

xy(x + y) + yz(y + z) + xz (z + x)

x²y + y²x + y²z + z²y + xz² + x²z

xyz(x + y + y + z + z + x)

xyz(2x + 2y + 2z)

2xyz(x + y + z)

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Answer 2

$\large\underline{\sf{Solution-}}$

Given that

$\red{\rm :\longmapsto\:x, \: y, \: z \: > \: 0}$

and

$\rm :\longmapsto\:\dfrac{x + y}{z} + \dfrac{y + z}{x} + \dfrac{z + x}{y}$

$\rm \:  =  \: \dfrac{x}{z} + \dfrac{y}{z} + \dfrac{y}{x} + \dfrac{z}{x} + \dfrac{z}{y} + \dfrac{x}{y}$

can be re-arranged as

$\rm \:  =  \: \bigg[\dfrac{x}{y} + \dfrac{y}{x} \bigg] + \bigg[\dfrac{y}{z} + \dfrac{z}{y} \bigg] + \bigg[\dfrac{z}{x} + \dfrac{x}{z} \bigg]$

We know,

If a and b are two real positive numbers, then Arithmetic Mean and Geometric Mean between a and b are related as

$\boxed{ \tt{ \: AM \geqslant GM\rm \implies\:a + b \geqslant 2 \sqrt{ab} \: }}$

So, using this, in above expression, we get

$\rm \: \geqslant   \: 2 \sqrt{\dfrac{x}{y} \times \dfrac{y}{x} } + 2 \sqrt{\dfrac{y}{z} \times \dfrac{z}{y} } + 2 \sqrt{\dfrac{z}{x} \times \dfrac{x}{z} }$

$\rm \:  \geqslant   \: 2 \sqrt{1} + 2 \sqrt{1} + 2 \sqrt{1}$

$\rm \:  \geqslant   \: 2 + 2 + 2$

$\rm \:  \geqslant   \: 6$

Hence,

$\rm :\longmapsto\:\dfrac{x + y}{z} + \dfrac{y + z}{x} + \dfrac{z + x}{y} \geqslant 6$

So, Minimum value

$\rm :\longmapsto\:\dfrac{x + y}{z} + \dfrac{y + z}{x} + \dfrac{z + x}{y} = 6$

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1. Arithmetic mean between two positive numbers x and y is given by

$\red{\rm :\longmapsto\:\boxed{ \tt{ \: AM = \frac{x + y}{2} \: }}}$

2. Geometric mean between two positive real numbers x and y is given by

$\red{\rm :\longmapsto\:\boxed{ \tt{ \: GM = \sqrt{xy} \: }}}$

3. Harmonic mean between two positive real numbers x and y is given by

$\red{\rm :\longmapsto\:\boxed{ \tt{ \: HM = \frac{2xy}{x + y} \: }}}$

4. Relationship between Arithmetic mean, Geometric mean and Harmonic mean

$\red{\rm :\longmapsto\:\boxed{ \tt{ \: AM \geqslant GM \geqslant HM \: }}}$

$\red{\rm :\longmapsto\:\boxed{ \tt{ \: {GM}^{2} \: = \: AM \: \times \: HM \: }}}$