How can I prove that x−x22<ln⁡(1+x)

PEEWSRIGWETRYqx

PEEWSRIGWETRYqx

Answered

2022-01-21

How can I prove that xx22<ln(1+x)

Answer & Explanation

Linda Birchfield

Linda Birchfield

Expert

2022-01-21Added 39 answers

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!
Maria Lopez

Maria Lopez

Expert

2022-01-22Added 32 answers

consider f(x)=ln(1+x)x+x22
f(x)=x21+x>0  forall  x>0
Hence f(x)>f(0)

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