Question

Use the sign of the derivative to establish the inequaltiy ln(1 + x) > x − x 2 2 , ∀ x > 0

Answer 1

Define function $f(x)$ as:

$f(x)=\frac{x^2}{2}+ln(1+x)-x$

Note that: $f(0)=0$

Let us find first derivative of function $f(x)$ :

$f'(x)=\frac{x^2}{x+1}\ ,$

Note that $f'(x)>0$ for all $x>0$

Since $f(0)=0$ and $f'(x)>0$ for all $x>0$ it follows that:

$\frac{x^2}{2}+ln(1+x)-x>0\Rightarrow ln(1+x)>x-\frac{x^2}2{}$ , for all $x>0$

Hence Proved.