prove x-(x+1)ln(x+1)<0 if x>0 in the Inequality

Expert Answers
sciencesolve eNotes educator| Certified Educator

You should use derivative of the function to predict if the function decreases over `(0,oo)`  such that:

`f'(x) = 1 - (x+1)'*ln(x+1) - (x+1)*(ln(x+1))'`

`f'(x) = 1 - ln(x+1) - (x+1)*(1/(x+1))`

`f'(x) = 1 - ln(x+1) - 1`

Reducing like terms yields:

`f'(x) = - ln(x+1)`

Notice that for all positive x, the value of derivative is negative, hence, the function decreases over `(0,oo)` , thus `x-(x+1)ln(x+1)lt` 0 if `x in (0,oo).`