<s>[INST] Validating the inequality. $x-\frac{x^2}{2}+\frac{x^3}{3(1+x)}<\log(1+x)<x-\frac{x^2}{2}+\frac{x^3}{3}$, $x>0$
Here I can only see that the right side of second inequality i.e. $x-\frac{x^2}{2}+\frac{x^3}{3}$ comes in the expansion of $\log(1+x)$.
We have done the Lagrange's mean value theorem and intermediate value theorem, do these have anything to do with the inequality.
Kindly provide some hint.
 [/INST] defining $$f(x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\ln(1+x)$$ for $x=0$ we get $$f(0)=0$$ and $$f'(x)=1-x+x^2-\frac{1}{x+1}=\frac{(1-x+x^2)(1+x)-1}{x+1}=\frac{x^3}{1+x}>0$$ for $x>0$ thus our function is monotonously increasing and we get $$f(x)>0$$ for all $$x>0$$
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