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Let `f:""(-1,""1) rarr R` be a differentiable function with `f(0)""=""-1" "a n d" "f'(0)""=""1` . Let `g(x)""=""[f(2f(x)""+""2)]^2` . Then `g'(0)""=` (1) `-4` (2) 0 (3) `2` (4) 4

A

0

B

`-2`

C

4

D

`-4`

Text Solution

Verified by Experts

The correct Answer is:
D
Let `g(x)={f(2 f(x)+2}^(2).Then`
`g'(x)=2f(2f(x)+2)xx(d)/(dx){f(2f(x)+2}`
`Rightarrow g'(x)=2f(2f(x)+2)xxf'(2f(x))xx(d)/(dx)(2f(x)+2)`
`Rightarrow g'(x)=2f(2f(x)+2)xxf'(2f(x)+2)xx2f'(x)`
Putting x=0, we get
`g'(0)=2f(2f(0)+2)xxf'(2f(0)+2)xx2f'(0)`
`Rightarrow g'(0)=2f(2f(0)xxf'(0)xx2f'(0)`
`Rightarrow g'(0)=2xx1xx2xx1=-4`
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