Show that- $$x-\frac{x^2}{2}<\ln(1+x)<x-\frac{x^2}{2(1+x)}, x>0$$
I can prove this by observing that at $x=0$, the three functions in the above inequality (say $f_1<f_2<f_3$) are zero. And for $x>0$, the derivative of the difference of $f'_2-f'_1>0$ and $f'_3-f'_2>0$ therefore the inequality is satisfied.
I am looking for an elegant solution to this using some kind of Mean Value Theorem maybe, if possible.
Well, for any $x>0$ we have $\log(1+x)=\int_{0}^{x}\frac{dt}{1+t}$, where $\frac{1}{1+t}>1-t$ is trivial and $$ \int_{0}^{x}\frac{dt}{1+t} = x-\int_{0}^{x}\frac{t}{t+1}\,dt<x-\int_{0}^{x}\frac{t}{x+1}\,dt = x-\frac{x^2}{2(x+1)}. $$