Let f : ℝ → ℝ satisfy f x + y = 2 x f y + 4 y ( f x , ∀ x , y ∈ ℝ . If f…

Given, $f\left(x+y\right)=2^{x}f\left(y\right)+4^y(f\left(x\right)$

Put $y=2$ we get,

$f\left(x+2\right)=2^x\times 3+16f\left(x\right)$

$f^{\text{'}}\left(x+2\right)=16f^{\text{'}}\left(x\right)+3\times 2^x\ln 2$

Now put $x=2$ we get,

$f^{\text{'}}\left(4\right)=16f^{\text{'}}\left(2\right)+12\ln 2 \dots \left(\mathrm{i}\right)$

Similarly, $f\left(y+2\right)=4f\left(y\right)+3\times 4^y$

$f^{\text{'}}\left(4\right)=4f^{\text{'}}\left(y\right)+3\times 4^y\ln 4$

$f^{\text{'}}\left(4\right)=4f^{\text{'}}\left(2\right)+96\ln 2 \cdots \left(\mathrm{ii}\right)$

solving eq. $\left(\mathrm{i}\right)$ and $\left(\mathrm{ii}\right)$, we get

$f^{\text{'}}\left(2\right)=7\ln 2$

from equation $\left(\mathrm{i}\right)$, we get

$f^{\text{'}}\left(4\right)=124\ln 2$

Now $\Rightarrow 14\times \frac{f^{\text{'}}\left(4\right)}{f^{\text{'}}\left(2\right)}$

$14\times \frac{124\ln 2}{7\ln 2}=248$