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

Options

A. `f(x)gte^(2x)in (0,oo)`
B. f(X) is decreasing in `(0,oo)`
C. f(X) is increasing in `(0,oo)`
D. `f'(x)lte^(2x)`in `(0,oo)`

Answer

1,3

Explanation

Give that `f(x)gt2f(x)forall x in R`
`rarr f(x)-2f(x)gt0 forall x in R`
`therefore e^(-2x)f(x)-2f(x)gt0 forall x in R`
`rarr (d)/(dx)(e^(-2x)f(x))gt0 forall x in R`
Let g(x)=`e^(-2x)f(x)`
Now `g(X) gt 0 forall x in R`
Also g(0)=1
`rarr g(X)gt(0)=1`
`rarr f(x)gte^(2x)forall x in (0,oo)`
So option 1 is correct
f(X) is strictly increasing on x in `(0,oo)`
so option 3 is correct
Clearly option 4 is incorrect