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The set of values for which (log)e(1+x)l...

The set of values for which `(log)_e(1+x)lt=x` is equal to_______

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The correct Answer is:
`x in (-(1)/(2), 0) uu ((1)/(2) ,oo) , x in (-oo, - (1)/(2)) uu (0, (1)/(2))`
Let `f(x) = log (1 + x ) - x`
`rArr " " f ' (x) = (1)/( 1 + x) - 1 = - (x) /( 1 + x )`
`rArr " " f ' (x) gt 0`
when ` " " - 1 lt x lt 0`
and ` " " f ' (x) lt 0`
when `" " x gt 0 `
`therefore f (x)` in increasing for ` - 1 lt x lt 0`
` rArr " " f(x) lt f (0)`
`rArr " " log (1 + x ) lt x `
Again ` f(x) ` is decreasing for `x gt 0`
`rArr " "f (x) lt f(0)`
`rArr " " log (1 + x ) lt x `
`therefore " " log (1 + x) le x , AA x gt -1 `
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