Home
Class 12
MATHS
If f:R is a differentiable fucntion suc...

If f:R is a differentiable fucntion such that f(x) `gt` 2f(x) for all x `in` R 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 `
Promotional Banner

Similar Questions

Explore conceptually related problems

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

Let f:(0,oo)->R be a differentiable function such that f'(x)=2-f(x)/x for all x in (0,oo) and f(1)=1 , then

If f:R->R is a twice differentiable function such that f''(x) > 0 for all x in R, and f(1/2)=1/2. f(1)=1, then

Let f be a differentiable function such that f(1) = 2 and f'(x) = f (x) for all x in R . If h(x)=f(f(x)), then h'(1) is equal to

Let f: R->R be a twice differentiable function such that f(x+pi)=f(x) and f''(x)+f(x)geq0 for all x in Rdot Show that f(x)geq0 for all x in Rdot

Let f:R to R and h:R to R be differentiable functions such that f(x)=x^(3)+3x+2,g(f(x))=x and h(g(x))=x for all x in R . Then,

Let f be a twice differentiable function such that f''(x)gt 0 AA x in R . Let h(x) is defined by h(x)=f(sin^(2)x)+f(cos^(2)x) where |x|lt (pi)/(2) . The number of critical points of h(x) are

Let f:R->R be a continuous function such that |f(x)-f(y)|>=|x-y| for all x,y in R ,then f(x) will be

If f(x) is a differentiable real valued function satisfying f''(x)-3f'(x) gt 3 AA x ge 0 and f'(0)=-1, then f(x)+x AA x gt 0 is

Let f defined on [0, 1] be twice differentiable such that | f"(x)| <=1 for x in [0,1] . if f(0)=f(1) then show that |f'(x)<1 for all x in [0,1] .