Mathematics - Functions Question with Solution | TestHub

$P\left(x,y\right): f\left(x+y\right)=f\left(x\right)f^{\text{'}}\left(y\right)+f^{\text{'}}\left(x\right)f\left(y\right) \forall x, y\in R$
$P\left(\mathrm{0,0}\right): f\left(0\right)=f\left(0\right)f^{\text{'}}\left(0\right)+f^{\text{'}}\left(0\right)f\left(0\right)$
$\Rightarrow 1=2f^{\text{'}}\left(0\right)$
$\Rightarrow f^{\text{'}}\left(0\right)=\frac{1}{2}$
$P\left(x,0\right): f\left(x\right)=f\left(x\right).f^{\text{'}}\left(0\right)+f^{\text{'}}\left(x\right).f\left(0\right)$
$\Rightarrow f\left(x\right)=\frac{1}{2}f\left(x\right)+f^{\text{'}}\left(x\right)$
$\Rightarrow f^{\text{'}}\left(x\right)=\frac{1}{2}f\left(x\right)$
$\Rightarrow f\left(x\right)=e^{\frac{1}{2}x}$
$\Rightarrow \ln \left(f\left(4\right)\right)=2$