**Relationship among different means: AM≥GM≥HM**

where, A.M=Arithmatic mean, G.M=Geomatric mean and H.M=Harmonic mean

**💎 Proof**:

Let us consider two observation *x*_{1} and *x*_{2}. Then

i,e. AM>GM.

The equality sign holds good, if

*x*

_{1}=

*x*

_{2}.

Again

i,e. GM>HM

The equality sign holds good, if

*x*

_{1}=

*x*

_{2}.

∴ AM≥GM≥HM

The theorem is true for any number of observations.

**Relationship Between A.M. and GM.**

Let *A* and *G* be A.M. and G.M. of two given positive real numbers *a* and *b*, respectively. Then

From (i), we obtain the relationship A≥G.

📌 Example 1. If A.M. and GM. of two positive numbers *a* and *b* are 10 and 8, respectively, find the numbers.

✍ Solution:

Given that A.M. =½(*a*+*b*)=10…(1)

and G, M, =√(*a*⋅*b*)=8…(2)

Equations (1) and (2) can be written as

*a*+

*b*=20…(1)

*a*⋅

*b*=64…(2)

Putting the value of

*a*and

*b*from (1), (2) in the identity (

*a*–

*b*)

^{2}=(

*a*+

*b*)

^{2}-4

*ab*, we get

(

*a*–

*b*)

^{2}=400-256=144

or

*a*–

*b*=±12 … (3)

Solving (1) and (3), we obtain

*a*=4,

*b*=16 or

*a*=16,

*b*=4

Thus, the numbers

*a*and

*b*are 4, 16 or 16, 4 respectively.

📌 Ex2. The A.M between two numbers is 10 and their G.M is 8. Determine the numbers.

✍ Solution:

A.M =½(*a*+*b*)=10→*a*+*b*=20…(1)

G.M. =√*a*⋅*b*=8→*a*⋅*b*=64…(2)

from (2), *b*=64/*a*, put in (1)

Hence the numbers are 4 and 16.

📌 Ex3. If A.M. and G.M. of roots of a quadratic equation are 8 and 5, respectively, then obtain the quadratic equation.

✍ Solution:

Let the root of the quadratic equation be *a* and *b*.

According to the given condition,

A.M. ½(*a*+*b*)=8⇒*a*+*b*=16…(1)

G.M. √*a*⋅*b*=5⇒*a*⋅*b*=25…(2)

The quadratic equation is given by,

*x*

^{2}–

*x*(Sum of roots)+(Product of roots)=0

*x*

^{2}–

*x*(

*a*+

*b*)+(ab)=0

*x*

^{2}-16

*x*+25=0 [Using (1) and (2)]

Thus, the required quadratic equation is

*x*

^{2}-16

*x*+25=0.

**Objective Type Questions**

Choose the correct answer out of the four given options in Examples 4 and 5.

📌 Example 4. If *x*, *y*, *z* are positive integers then value of expression (*x*+*y*)(*y*+*z*)(*z*+*x*) is

(A) =8*xyz* (B) >8*xyz* (C) <8*xyz* (D) =4*xyz*

✍ Solution: (B) is the correct answer, since A.M. > G.M.,

Multiplying the three inequalities, we get

📌 Example 5. The minimum Value of the expression 3^{x}+3^{(1-x)}, *x*∈*R*, is

(A) 0 (B) ⅓ (C) 3 (D) 2√3

✍ Solution: (D) is the correct answer. We know A.M.≥G.M. for positive numbers.

Therefore,