inequality proving problem

hearts123

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Joined
Feb 22, 2019
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Hi everyone! I have a proving problem that I don't know how to do, at all. Any help is greatly appreciated!!

Given a and b as positive real numbers, and a is not equal to b
Prove that a^5 + b^5 > (a^3)(b^2) + (a^2)(b^3)
 

pka

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Jan 29, 2005
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8,559
Given a and b as positive real numbers, and a is not equal to b
Prove that \(\displaystyle a^5 + b^5 > (a^3)(b^2) + (a^2)(b^3)\)
\(\displaystyle \begin{align*}(a+b)^5 &=a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5\\ &\ge5a^4b+10a^3b^2+10a^2b^3+5ab^4 \\&\ge 10a^3b^2+10a^2b^3\\& \text{Can you finish?} \end{align*}\)
 
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