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prove x-(x+1)ln(x+1)<0 if x>0 in the Inequality
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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).`
Posted by sciencesolve on June 4, 2012 at 3:17 PM (Answer #1)
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