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Let f(x) =int1^x e^t/tdt,x in R^+ . The...

Let `f(x) =int_1^x e^t/tdt,x in R^+` . Then complete set of valuesof x for which `f(x) leq In` x is

A

`(0,1]`

B

`[1,oo)`

C

`(0,oo)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A
`f(x)=int_(1)^(x) (e^(t))/tdtimpliesf(1)=0` and `f'(x)=(e^(x))/x`
Let `g(x)=f(x)-In (x),x epsilonR^(+)`
`:.g'(x)=f'(x)-1/x=(e^(x)-1)/x gt 0 AA x epsilonR^(+)`
Thus `g(x)` is increasing for `x epsilonR^(+)`
`g(1)=f(1)-In1=0-0=0`
`:. g(x)gtAAxgt1` and `g(x)le0AAxepsilon(0,1]`
`:.In x ge f(x)AA x epsilon (0,1]`
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