QJEEM10Mac8e2: JEE Main 2010 Mathematics - Class 12 JEEM10Mac8e2 Solution: Let be a differentiable function with and . Let . Then

Question

Verified Solution • 2025–26 Session moderate

Let

f:\left( { - 1,1} \right) \to R

be a differentiable function with

f\left( 0 \right) = - 1

and

f'\left( 0 \right) = 1

. Let

g\left( x \right) = {\left[ {f\left( {2f\left( x \right) + 2} \right)} \right]^2}

. Then

g'\left( 0 \right) =

Accepted Answer

g'\left( x \right) = 2\left( {f\left( {2f\left( x \right) + 2} \right)} \right)

\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{d \over {dx}}\left( {f\left( {2f\left( x \right) + 2} \right)} \right)} \right)

= 2f\left( {2f\left( x \right) + 2} \right)f'\left( {2f\left( x \right)} \right)

\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left. 2 \right).\left( {2f'\left( x \right)} \right)

\Rightarrow g'\left( 0 \right) = 2f\left( {2f\left( 0 \right) + 2} \right).

\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,f'\left( {2f\left( 0 \right) + 2} \right).2f'\left( 0 \right)

= 4f\left( 0 \right){\left( {f'\left( 0 \right)} \right)^2}

= 4\left( { - 1} \right){\left( 1 \right)^2} = - 4

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