If f:R is a differentiable fucntion such that f(x) `gt` 2f(x) for all x `in` R and f(0) =1 then

Options

A. f(x) is incresing in `(0,oo)`
B. f(x) is decereasing in `(0,oo)`
C. `f(x) lt e^(2x) " in " (0,oo)`
D. `f(x)lt e^(2x) in (0,oo)`

Answer

A, C

Explanation

We have e,
`f(x) lt 2 f(x) " "for all x in R `
`rArr f(x) -2 f(x) gt 0 " " for all x in R `
`rArr f(x) e^(-2x)-2e^(-2x) f(x) gt 0" " for all x in R`
`rArr d/dx ( f(x) e^(-2x))gt 0 " "for all x in R `
`rArr f(x)e^(-2x) ` is an increasing on R
`rArr f(x) e^(-2x) gt f(0)e^0 " " for all x gt 0 `
`rArr f(x) gt e^(2x)`
Now
`f(x) gt 2 f(x)" "["Given"]`
`rArr f(x) gt 2 e^(2x)" "[because f(x) gt e^(2x)]`
`rArr f(x) gt 0 " for all " x gt 0 `
`rArr ` f(x) is increasing in`(0,oo)`
Hence options (a) and (c) are true.