KVPY Sample Paper KVPY Stream-SX Model Paper-9

If \[f:R\to R\] be a differentiable function such that \[f(x+2y)=f(x)+\,f(2y)+4xy\forall \,x,y\in R\] than;

A) \[f'(1)=\,f'(0)+1\]

B) \[f'(1)=\,f'(0)-1\]

C) \[f'(0)=\,f'(1)+2\]

D) \[f'(0)=\,f'(1)-2\]

Correct Answer: D

Solution :

Let \[x=y=0\Rightarrow f(0)=0\]

Now we have\[\begin{align} & \frac{f\left( x+2y \right)-f(x)}{2y}=2x+\frac{f\left( 2y \right)}{2y} \\ & =2x+\frac{f(2y)-f(0)}{2y} \\ \end{align}\]

Taking limit on both sides as \[~2y\to 0\]we get

\[f'(x)=2x+f'(0)\,as\,f(0)=0\]

\[\Rightarrow f'(0)=f'(1)-2\]