Given that `f(x) gt f(x) for all x in R` and f(0) =g(0) then

Options

A. `f(x) gt g (x) for all x in (0,oo) and f(x) gt g(x) for all x in (-oo,0)`
B. `f(x)lt g(x) for all x in (0,oo) and f(x) gtg(x) for all x in (-oo,0)`
C. `f(x) gt g (x) gt for all x in (- oo,0)and f(x) lt g(x) for all x in (0,oo)`
D. none of these

Answer

A

Explanation

Let `h(x) =f(x) -g(x) for all x in R` Then
`h (x) =f'(x)-g' (x) for all x in R`
`rArr h(x) gt 0 for all x In R `
`rArr f(x) ` is increasing on R.
But, h(0) =f(0)-g(0) =0
`therefore h(x) lt h (0) for all x gt 0 `
`rArr f(x)-g(x) gt 0 for all x in (- oo ,0)`
`rArr f(x) gt g(x) for all x in (0,oo)`
and
`f(x) lt g (x) for all x in (-oo, 0)`