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

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gnollgnoll | Student, Undergraduate | (Level 1) eNoter

Posted June 4, 2012 at 3:04 PM via web

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

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted June 4, 2012 at 3:17 PM (Answer #1)

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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).`

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