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If f:RR-> RR is a differentiable funct...

If ` f:RR-> RR ` is a differentiable function such that `f(x) > 2f(x) ` for all `x in RR ` and `f(0)=1, ` then

A

`f(x) gt e^(2x)` in `(0, oo)`

B

`f'(x) lt e^(2x)` in `(0, oo)`

C

`f(x)` is increasing in `(0, oo)`

D

`f(x)` is decreasing in `(0,oo)`

Text Solution

Verified by Experts

The correct Answer is:
C, D
`f' (x) gt 2 f(x) rArr (dy )/(y) gt 2 dx `
`rArr " " overset(f(x)) underset(1)int (dy)/(y) gt 2 overset x underset 0 dx `
`" " "In " (f (x)) gt 2x `
`therefore " " f(x) gt e ^(2 x )`
Also, as ` f ' (x) gt 2 f(x)`
`therefore f' (x) gt 2 e ^(2x) gt 0 `
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