Analysis Homework. Proof of Arithmetic-Means Inequality

[The_Iceflash]

From the Text, Introduction to Analysis, by Arthur Mattuck pg 32 2-3 (a)

Homework Statement

Prove: for any a,b \geq 0, \sqrt{ab} \leq \frac{\left(a+b\right)}{2}
with equality holding if and only if a =b

Homework Equations

All Perfect squares are \geq 0

The Attempt at a Solution

I wasn't sure where to go with this but I took the inequality:
\sqrt{ab} \leq \frac{\left(a+b\right)}{2}

and squared both sides to give me:

ab \leq \frac{\left(a+b\right)^{2}}{4}

I then separated the right which gave me:

ab \leq \frac{a^{2}}{4} + \frac{ab}{2} + \frac{b^{2}}{4}

I multiplied both sides by 4 which gave me:

4ab \leq a^{2} + 2ab + b^{2}

I quickly saw that 4ab \leq \left(a+b\right)^{2}

From the known equation: 0 \leq \left(a+b\right)^{2}

I added the two together and got: 4ab \leq2 \left(a+b\right)^{2}

which divided by 4 equals: ab \leq \frac{\left(a+b\right)^{2}}{2}

This as far as I got. Help would be appreciated.

 

[Mark44]

See this thread: https://www.physicsforums.com/showthread.php?t=376036

 

[The_Iceflash]

See this thread: https://www.physicsforums.com/showthread.php?t=376036

Thanks.