Which of the following options is the only INCORRECT combination ?

Answer

C

Explanation

`f(x) = x + `In `x - x" In "x`
`f (1) = 1 gt 0`
`f(e^(2)) = e ^(2) + 2 - 2e ^(2) = 2 - e ^(2) lt 0`
`rArr f(x) = 0 ` for some ` x in (1, e ^(2))`
`therefore ` I is correct
`f'(x) = 1 + (1)/(x)- In x - 1 = (1)/(x) - In x `
`f ' (x) gt 0 `for ` (0, 1)`
`f' (x) lt 0` for ` ( e, oo)`
`therefore P and Q` are correct , II is correct, III is incorrect.
`f '' (x) = (-1)/(x ^(2)) - (1)/(x)`
`therefore S`, is correct, R is incorrect.
IV is incorrect.
`underset(x to oo) (lim) f(x) = -oo `
` underset (x to oo ) (lim) f ' (x) = -oo `
` underset (x to oo) (lim) f '' (x) = 0`
`therefore ` ii, iii, iv are correct.