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)
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