From the Text, Introduction to Analysis, by Arthur Mattuck pg 32 2-3 (a)(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

Prove: for any a,b [tex]\geq[/tex] 0, [tex]\sqrt{ab}[/tex] [tex]\leq[/tex] [tex]\frac{\left(a+b\right)}{2}[/tex]

with equality holding if and only if a =b

2. Relevant equations

All Perfect squares are [tex]\geq[/tex] 0

3. The attempt at a solution

I wasn't sure where to go with this but I took the inequality:

[tex]\sqrt{ab}[/tex] [tex]\leq[/tex] [tex]\frac{\left(a+b\right)}{2}[/tex]

and squared both sides to give me:

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

I then separated the right which gave me:

ab [tex]\leq[/tex] [tex]\frac{a^{2}}{4}[/tex] + [tex]\frac{ab}{2}[/tex] + [tex]\frac{b^{2}}{4}[/tex]

I multiplied both sides by 4 which gave me:

4ab [tex]\leq[/tex] [tex]a^{2}[/tex] + [tex]2ab[/tex] + [tex]b^{2}[/tex]

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

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

I added the two together and got: 4ab [tex]\leq[/tex]2 [tex]\left(a+b\right)^{2}[/tex]

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

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

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# Homework Help: Analysis Homework. Proof of Arithmetic-Means Inequality

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