# A question on derivation...

#### kkafal

##### New member
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

##### New member
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

##### Well-known member
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

$\chi$ $\sigma$