JEE Advanced2017MathematicsDifferential EquationsActual
If f : R → R is a differentiable function such that f ′ x > 2 f ( x ) for all x ∈ R and f 0 = 1 then
Options
- Af x > e 2 x i n ( 0 , ∞ )
- Bf x i s d e c r e a s i n g i n ( 0 , ∞ )
- Cf x i s i n c r e a s i n g i n ( 0 , ∞ )
- Df   ' ( x ) < e 2 x   in   ( 0 , ∞ )
Correct answer
A. f x > e 2 x i n ( 0 , ∞ )
Step-by-step solution
Given that, f ' x > 2   f x   ∀   x ∈ R ⇒         f ' x - 2 f x > 0       ∀   x ∈ R ∴         e - 2 x   f ' x - 2 f x > 0   ∀   x ∈ R ⇒       d d x e - 2 x   f x > 0   ∀   x ∈ R Let g x = e - 2 x   f x Now, g ' x > 0   ∀   x ∈ R ⇒         g x is