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The_Iceflash
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From the Text, Introduction to Analysis, by Arthur Mattuck pg 32 2-3 (a)
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
All Perfect squares are [tex]\geq[/tex] 0
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.
Homework Statement
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
Homework Equations
All Perfect squares are [tex]\geq[/tex] 0
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.