Let f: ℝ → ℝ satisfy f(x + y) = 2 ^ x f(y) + 4 ^ y ( f(x), x , y ∈...

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

Put $y = 2$ we get,

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

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

Now put $x = 2$ we get,

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

Similarly, $f\left(y + 2\right) = 4f$\left(y\right) + 3 \times {4}^{y}$</p><p>${f}^{'}\left(4\right) = 4{f}^{'}\left(y\right) + 3 \times {4}^{y}\ln 4$</p><p>${f}^{'}\left(4\right) = 4{f}^{'}\left(2\right) + 96\ln 2 ⋯ \left(ii\right)$</p><p>solving eq. \left(i\right)$ and \left(ii\right)$, we get</p><p>${f}^{'}\left(2\right) = 7\ln 2$</p><p>from equation \left(i\right)$, we get

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

Now $⇒ 14 \times \dfrac{{f}^{'}\left(4\right)}{{f}^{'}\left(2\right)}$

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