1. ## Equation

Hi people,

I must show that: $\displaystyle \forall x \in \mathbb{R}: Arctanx>\frac{x}{1+x^2}$

I don't know how to do it???? can you help me please???

2. Originally Posted by bhitroofen01
Hi people,

I must show that: $\displaystyle \forall x \in \mathbb{R}: Arctanx>\frac{x}{1+x^2}$

I don't know how to do it???? can you help me please???
It's only true for $\displaystyle x>0$.

To prove this, show $\displaystyle \tan^{-1}(0)=\frac{0}{0^2+1}$.

Then look at the derivatives.

$\displaystyle \frac{d}{dx}\left(\tan^{-1}(x)\right) = \frac{1}{x^2+1}$

$\displaystyle \frac{d}{dx}\left(\frac{x}{x^2+1}\right) = \frac{1}{x^2+1}-\frac{4x^2}{(x^2+1)^2}$

What does this tell you?