Question

If a and b two different positive real numbers, then which of the following relation is true
A.$2\sqrt {ab} > \left( {a + b} \right)$
B.$2\sqrt {ab} < \left( {a + b} \right)$
C.$2\sqrt {ab} = \left( {a + b} \right)$
D.None of these

Answer 1

Hint: In order to find the relation between Arithmetic mean and Geometric mean, write the numerical equations for them and subtract the equation of GM from AM, solve the equations and find the range which will be\[AM \geqslant GM\], substitute the values and find the correct option.
Formula used:
$AM = \dfrac{1}{2}\left( {a + b} \right)$
$GM = \sqrt {ab} $

Complete step-by-step answer:
We are given two different real positive numbers a and b.
For the two numbers, we are finding the Arithmetic mean (AM), numerically written as:
$AM = \dfrac{1}{2}\left( {a + b} \right)$ ………(1)
Now, we are also finding the Geometric Mean (GM). Numerically written as:
$GM = \sqrt {ab} $ ………(2)
Subtracting equation 2 from 1:
$AM - GM = \dfrac{1}{2}\left( {a + b} \right) - \sqrt {ab} $
$ \Rightarrow AM - GM = \dfrac{{\left( {a + b} \right)}}{2} - \sqrt {ab} $
Multiplying and dividing$\sqrt {ab} $ by $2$:
$ \Rightarrow AM - GM = \dfrac{{\left( {a + b} \right)}}{2} - \dfrac{{2\sqrt {ab} }}{2}$
$ \Rightarrow AM - GM = \dfrac{{\left( {a + b} \right) - 2\sqrt {ab} }}{2}$
$ \Rightarrow AM - GM = \dfrac{{a + b - 2\sqrt {ab} }}{2}$
Since, we can write ${\left( {\sqrt a } \right)^2} = a$ and ${\left( {\sqrt b } \right)^2} = b$, so writing it in the above equation:
$ \Rightarrow AM - GM = \dfrac{{{{\left( {\sqrt a } \right)}^2} + {{\left( {\sqrt b } \right)}^2} - 2\sqrt {ab} }}{2}$
Since, we know that ${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$, so we can write ${\left( {\sqrt a } \right)^2} + {\left( {\sqrt b } \right)^2} - 2\sqrt {ab} $ as ${\left( {\sqrt a - \sqrt b } \right)^2}$.
$ \Rightarrow AM - GM = \dfrac{{{{\left( {\sqrt a - \sqrt b } \right)}^2}}}{2}$
We can see that for any value of a and b, we will get the value greater than zero.
Numerically written as:
$ \Rightarrow AM - GM = \dfrac{{{{\left( {\sqrt a - \sqrt b } \right)}^2}}}{2} \geqslant 0$
\[ \Rightarrow AM - GM \geqslant 0\]
Adding both sides by GM, we get:
\[ \Rightarrow AM - GM + GM \geqslant 0 + GM\]
\[ \Rightarrow AM \geqslant GM\]
Substituting the value of AM and GM from equation and equation 2, we get:
\[ \Rightarrow \dfrac{1}{2}\left( {a + b} \right) \geqslant \sqrt {ab} \]
Multiplying both sides by 2:
\[ \Rightarrow \dfrac{1}{2} \times 2\left( {a + b} \right) \geqslant 2\sqrt {ab} \]
\[ \Rightarrow \left( {a + b} \right) \geqslant 2\sqrt {ab} \]
\[ \Rightarrow 2\sqrt {ab} \leqslant \left( {a + b} \right)\]
And, it matches with Option 2;
So, the correct answer is “Option 2”.

Note: Arithmetic Mean can be found out by adding the two real positive numbers and dividing them by 2.
Geometric mean is found out by taking the square root of the product of two numbers.
It should always be remembered that \[AM \geqslant GM\] for two positive real numbers.