Prove by Mean Value Theorem x1+x<ln⁡(1+x)<x for x>0

siutulysr5

siutulysr5

Answered

2022-01-21

Prove by Mean Value Theorem x1+x<ln(1+x)<x  for  x>0

Answer & Explanation

mihady54

mihady54

Expert

2022-01-22Added 13 answers

By the mean value theorem, given x>0, there exists c(0,x) such that f(x)f(0)=f(c)x,  i.e.,  ln(1+x)=x1+c. Since 0<c<x,
11+x<11+c<1. Therefore x1+x<x1+c<x, i.e.,
x1+x<ln(1+x)<x.

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