If `f(x) = underset(0)overset(x)inte^(t^(2)) (t-2) (t-3) dt` for all `x in (0, oo)` , then

Options

A. `e^x lt 1 +x `
B. `log_e (1+x) lt x `
C. `sin x gt x `
D. `log x gt x `

Answer

B

Explanation

Consider the function
`f(x)=e^x-(1 +x)`
`rArr f(x)=e^x-1`
`f(x) lt 0 for x in (0,1)`
`rArr e^x -(1+x) gt "for all " x in (0,1)`
`rArr e^x + x " for all " x in (0,1)`
`rArr e^x gt 1 + x " for all" x in (0,1)`
Thus option (a) is not correct
`Let g(x)=log (1+x)-x ,x in (0,1) Then`

`g(x) lt g (0) "for all" x in (0,1)`
`rArr log_e (1)/(1+x) -1 = -(x)/(1+x) lt 0 " for all " (0,1)`
`rArr` g(x) is decreasing on (0,1)
`g(x) lt g (0) " for all " x in (0, 1)`
Thus option (b) is correct
Similarly it can be shown that options ( c) and (d) do not hold