How to prove that the function $$y(x)=x(\ln(x+1) - \ln(x))$$ is increasing on $[0,1]$? The derivative test requires to analyze equally challenging function $\ln{\left(\frac{x+1}{x}\right)}-\frac{1}{x+1}.$ Are there more ways to prove that $y(x)$ is strictly increasing?
Asked
Active
Viewed 64 times
2
-
Use $\log(\frac{x+1}{x})=-\log(1-\frac{1}{x+1})=\sum_{k\geq 1}\frac{1}{k(x+1)^k}$. – Myunghyun Song Nov 14 '18 at 11:17
1 Answers
1
Write your derivative as $$\ln{\left(1+\frac1x\right)}-\frac{\frac1x}{1+\frac1x}$$ and use the inequality $$\ln{(1+x)}>\frac{x}{1+x}$$ for $0<x<1$.
Edit: By definition $$\ln{(1+x)}:=\int_{1}^{1+x}\frac1tdt>\frac1{1+x}\int_{1}^{1+x}dt=\frac{x}{1+x}$$ where the inequality in the middle comes from the fact that $1/t$ is strictly decreasing.
Jimmy R.
- 35,868