Equation

**bhitroofen01** · April 3rd 2010, 10:02 AM

Hi people,

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

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

**chiph588@** · April 3rd 2010, 10:09 AM

Originally Posted by bhitroofen01

Hi people,

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

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

It's only true for $x>0$ .

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

Then look at the derivatives.

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

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

What does this tell you?

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