QJEEM22Ma4f0a: JEE Main 2022 Mathematics - Class 12 JEEM22Ma4f0a Solution: Let f : R R satisfy , x, y R. If f(2) = 3, then is equal to…

$\because$ $f(x + y) = {2^x}f(y) + {4^y}f(x)$ ....... (1)

Now, $f(y + x){2^y}f(x) + {4^x}f(y)$ ...... (2)

$\therefore$ ${2^x}f(y) + {4^y}f(x) = {2^y}f(x) + {4^x}f(y)$

$({4^y} - {2^y})f(x) = ({4^x} - {2^x})f(y)$

${{f(x)} \over {{4^x} - {2^x}}} = {{f(y)} \over {{4^y} - {2^y}}} = k$

$\therefore$ $f(x) = k({4^x} - {2^x})$

$\because$ $f(2) = 3$ then $k = {1 \over 4}$

$\therefore$ $f(x) = {{{4^x} - {2^x}} \over 4}$

$\therefore$ $f'(x) = {{{4^x}\ln 4 - {2^x}\ln 2} \over 4}$

$f'(x) = {{({{2.4}^x} - {2^x})ln2} \over 4}$

$\therefore$ ${{f'(4)} \over {f'(2)}} = {{2.256 - 16} \over {2.16 - 4}}$

$\therefore$ $14{{f'(4)} \over {f'(2)}} = 248$