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Let f:(0,oo)toR be given by f(x)=int(1//...

Let `f:(0,oo)toR` be given by `f(x)=int_(1//x)^(x)e^(-(t+1/t))(dt)/t`, then

A

`f(x)` is monotonically increasing on `[1, oo)`

B

`f(x)` is monotocially decreasing on `[0, 1)`

C

`f(x) + f((1)/(x))= 0, AA x in (0, oo)`

D

`f(2^(x))` is an odd function of `x` on R

Text Solution

Verified by Experts

The correct Answer is:
A, C
Given, `f(x) = int_((1)/(x))^(x) ( e ^(-(t +(1)/(t)))) /( t ) dt `
`f' (x) = 1 * ( e^(- (x + (1) /(x)))) /( x ) - ((-1)/(x^(2))) (e ^(-((1)/(x) + x)))/( 1//x)`
`" " = ( e ^(-(x + (1)(x))))/(x) + (e ^(- (x + (1)/(x))))/(x) = ( 2e ^(- (x+ (1)/(x))))/( x)`
As `f ' (x) gt 0 , AA x in ( 0, oo)`
`rArr ` Option (a) is correct and option (b) is wrong.
Now, `f(x) + f((1)/(x)) = int_((1)/(x)) ^(x) (e^(- (t + (1)/(t))))/(t) dt + int _(x) ^(1//x) (e ^(- (t + (1)/(t))))/(t) dt `
Now, let ` g(x) = f (2 ^(x)) = int_(2 ^(-x))^(2 ^(x)) (e^(-(t + (1)/(t))))/( t) dt `
`" " g(-x) = f(2^(-x)) = int_(2^(x)) ^(2^(-x)) (e ^(-(t + (1)/(t))))/( t) dt = - g (x)`
` therefore f(2 ^(x))` is an odd function.
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