KVPY Sample Paper KVPY Stream-SX Model Paper-20

Let f be a differentiable function such that \[f(1)=2\] and \[f'(x)=f(x)\] for all \[x\in R.\] If \[h(x)=f(f(x)),\] then h' (1) is equal to:

A) \[2{{e}^{2}}\]

B) 4e

C) 2e

D) \[4{{e}^{2}}\]

Correct Answer: B

Solution :

\[\frac{f'(x)}{f(x)}=1\] \[\forall \,x\,\in \,R\]

Integrate and use f(1) = 2

\[f(x)=2{{e}^{x-1}}\]\[\Rightarrow \]\[f'(x)=2{{e}^{x-1}}\]

\[h(x)=f(x)\]\[\Rightarrow \]\[h'(x)=f'(x)=f'(f(x))f'(x)\]

\[h'(1)=f'(f(1))f(1)\]

\[=f'(2)f'(1)\]

= 2e.2 = 4e.