KVPY Sample Paper KVPY Stream-SX Model Paper-12

Let \[f:\mathbf{R}\to \mathbf{R}\] be a continuous function which satisfies \[f(x)=\int\limits_{0}^{x}{f(t)dt.}\] Then the value of \[f\left( \ln 5 \right)\]is

A) \[-2\]

B) 3

C) \[-1\]

D) 0

Correct Answer: D

Solution :

given that \[f(x)=\int_{0}^{x}{f(t)}dt\]

Clearly \[f(0)=0.\]

Also\[f'(x)=f(x)\] \[\Rightarrow \]\[\frac{f'(x)}{f(x)}=1\]

Integrating both sides with respect to x, we get

\[\int{\frac{f'(x)}{f(x)}dx}=\int{1\,\,dx}\] \[\Rightarrow \]In \[f(x)=x+\ln C\Rightarrow f(x)=C{{e}^{x}}\]

Now \[f(0)=0\Rightarrow C{{e}^{x}}=0\Rightarrow C=0\]

\[\therefore \]\[f(x)=0\,\forall \,x\Rightarrow f(\ln 5)=0\]