Let `f:[(1)/(2),1]in R`(the set of all real numbers) be a positive, non-constant and differentiable function such that `f'(x) lt 2f(x')` and `f((1)/(2))=1`. Then the value of `int_(1//2)^(1) f(x) dx` lies in the interval

Options

A. (2e-1,2e)
B. e-1,2e-1)
C. `((e-1)/(2),e-1)`
D. `(0,(e-1)/(2))`

Answer

D

Explanation

We have,
`f'(x)-2f(x) lt 0`
`rArr f'(x)e^(-2x)-2e^(-2x)f(x) lt 0e^(-2x)`
`rArr (d)/(dx)(f(x)e^(-2x)) lt 0`
`rArr f(x)e^(-2x)` is decreasing on `[1//2,1]`
`rArr f(x)e^(-2x) lt f((1)/(2))` for all `x in [1//2,1]`
`rArr f(x)e^(-2x) lt (1)/(e )` for all `x in [1//2,1]`
`rArr f(x) lt e^(-2x-1)` for all `x in [1//2,1]`
Thus, we have,
`0 lt f(x) lt e^(2x-1)` for all `x in [1//2,1]`
`rArr 0 lt overset(1)underset(1//2)int f(x)dx lt overset(1)underset(1//2)int e^(2x-1)`
`rArr 0 lt overset(1)underset(1//2)int f(x) dx lt [(1)/(2)e^(2x-1)]_(1//2)^(1)`
`rArr 0 lt overset(1)underset(1//2)int f(x) dx lt (e-1)/(2)`
`rArr 0 lt overset(1)underset(1//2)int f(x) dx in (0,(e-1)/(2))`