JEE Advanced | 2015Set, Relation and FunctionHard
Question
Let g : 
→ 
be a differentiable functions with g(0) = 0, g′(0) = 0 and g′(1) ≠ 0. Let 
and h(x) = e|x| for all x ∈
. Let (ƒoh)(x) denote ƒ(h(x)) and (hoƒ)(x) denote h(ƒ(x)). Then which of the following is(are) true ?
→ be a differentiable functions with g(0) = 0, g′(0) = 0 and g′(1) ≠ 0. Let and h(x) = e|x| for all x ∈
. Let (ƒoh)(x) denote ƒ(h(x)) and (hoƒ)(x) denote h(ƒ(x)). Then which of the following is(are) true ?Options
A.ƒ is differentiable at x = 0
B.h is differentiable at x = 0
C.ƒoh is differentiable at x = 0
D.h of is differentiable at x = 0
Solution
(A) f(x) = 
⇒ f′(x) =
so ′A′ is right
(B) h(x) =
⇒ h′(x) = 
h′(0+) = 1, h′(0-) = - 1, ∴ B is wrong
(C) f(h(x)) = g(h(x)) as h(x) > 0
z = g(e|x|) =
z′ =
z ′(0+) = g′(1), z′(0-) = - g′(1) and g′(1) # - g′(1)
so C is wrong
(D)
⇒ f′(x) =
so ′A′ is right
(B) h(x) =
⇒ h′(x) = h′(0+) = 1, h′(0-) = - 1, ∴ B is wrong
(C) f(h(x)) = g(h(x)) as h(x) > 0
z = g(e|x|) =
z′ =
z ′(0+) = g′(1), z′(0-) = - g′(1) and g′(1) # - g′(1)
so C is wrong
(D)
Create a free account to view solution
View Solution FreeMore Set, Relation and Function Questions
For real x, let f (x) = x3 + 5x + 1, then...Let R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on the set A = {3, 6, 9, 12} ...Let f : [0, 2] → R be a function which is continuous on [0, 2] and is differentiable on (0, 2) with f(0) = 1. Let ...A velocity m/s is resolved into two components along OA and OB making angles 30o and 45o respectively with the given vel...The mean of the data set comprising of 16 observations is 16. If one of the observation valued 16 is deleted and three n...