Wednesday, 30 April 2014

inequality - How can I prove that $x-{x^2over2}



How can I prove that xx22<ln(1+x) for any x>0




I think it's somehow related to Taylor expansion of natural logarithm, when:



ln(1+x)=xx22+x33



Can you please show me how? Thanks.


Answer



Hint:



Prove that ln(1+x)x+x22 is strictly increasing for x>0.




edit: to see why this isn't a complete proof, consider x21 for x>0. It's strictly increasing; does that show that x2>1? I hope not, because it's not true!


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