Derivative of f:R->R with f'(x)=[x^2/1+(x^2))], f(0)=0 and its Bounds on R

[kkafal]

f:R->R is differentiable and f'(x)=[x^2/1+(x^2))] and f(0)=0.
check whether 0<=f(x)<=x for all x belonging R.
Thanks.

 

[kkafal]

A question on applications of differentiation

Show that the equation y=c1+c2.cos(y) such that c1>0 and 0<c2<1 , has only one root not more than c1+c2.

 

[chisigma]

f:R->R is differentiable and f'(x)=[x^2/1+(x^2))] and f(0)=0.
check whether 0<=f(x)<=x for all x belonging R.
Thanks.

In term of ODE we can write [if I undestood correctly...] ...

$\displaystyle y^{\ '}= \frac{x^{2}}{1 + x^{2}}\ ,\ y(0)=0$ (1)

The (1) is an ODE with separable variables and has as solution...

$\displaystyle y= x - \tan^{-1} x$ (2)

... so that Yor relation is satisfied... but only for $x \ge 0$... Kind regards$\chi$ $\sigma$