1. ## Demonstration

Prove :

2. Hi

There might be a better way to do this, but I´ll give one solution.
Note: $x-\frac{x^{2}}{2}$ is the first two terms of the maclaurin series of $ln(1+x)$ .

Anyway, let $f(x)=x-\frac{x^{2}}{2}-ln(1+x)$ , then we want to show that $f(x)<0$ for all $x>0$.

Solution: $f$ is continous and differentiable for all $x>-1$ , with $f'(x)=\frac{-x^{2}}{1+x}$ . (I have already simplified the derivative here).

Note tha we are only interested in $x>0$, so we don´t have to worry about the denominator of the derivative becoming zero.

Looking at $f'(x)$ we see that the only stationary point is $x=0$ . And for all $x>0$ we have that $f'(x)<0$ .

And because $f(0)=0$ , and the function is strictly decreasing for all $x>0$ , it immediately follows that $f(x)<0$ .

Now for the right inequality:

You could do the same thing for $g(x)=x-ln(1+x)$ , and show that $g(x)>0$ .

3. Let $f(x)=\ln(1+x)-x, \ f0,\infty)\to\mathbf{R}" alt="f(x)=\ln(1+x)-x, \ f0,\infty)\to\mathbf{R}" />

$f'(x)=-\frac{x}{1+x}<0, \ \forall x>0$, so f is strictly decreasing.

Then, $x>0\Rightarrow f(x)

Try to prove the other inequality in a similar way by using the function

$g(x)=x-\frac{x^2}{2}-\ln(1+x), \ x>0$