Let $\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$ satisfy $\mathrm{f}(\mathrm{x}+\mathrm{y})=2^{\mathrm{x}} \mathrm{f}(\mathrm{y})+4^{\mathrm{y}} \mathrm{f}(\mathrm{x}), \forall \mathrm{x}$, $y \in \mathbb{R}$. If $f(2)=3$, then $14 \cdot \frac{f^{\prime}(4)}{f^{\prime}(2)}$ is equal to ______.

JEE Main · 2022 · 26 Jun 2022 · Shift 2 · Sets Relations and Functions

Let

f : R \to R

satisfy

f (x + y) = 2^{x} f (y) + 4^{y} f (x), \forall x

y \in R

. If

f (2) = 3

, then

14 \cdot \frac{f ^{'} ( 4 )}{f ^{'} ( 2 )}

is equal to ______.

Answer: 248

Putting

y = 2

in the functional equation:

f (x + 2) = 2^{x} \cdot 3 + 16 f (x)

.

Differentiating:

f^{'} (x + 2) = 3 \cdot 2^{x} ln 2 + 16 f^{'} (x)

.

At

x = 2

f^{'} (4) = 12 ln 2 + 16 f^{'} (2)

... (i)

Putting

x = 2

in the differentiated form with

y

as variable:

f^{'} (y + 2) = 4 f^{'} (y) + 3 \cdot 4^{y} ln 4

.

At

y = 2

f^{'} (4) = 4 f^{'} (2) + 96 ln 2

... (ii)

Solving (i) and (ii):

f^{'} (2) = 7 ln 2

and

f^{'} (4) = 124 ln 2

.

Therefore

14 \cdot \frac{f ^{'} ( 4 )}{f ^{'} ( 2 )} = 14 \cdot \frac{124 l n 2}{7 l n 2} = 248

.

Answer: 248

JEE Main · 2022 · 26 Jun 2022 · Shift 2 · Integer answer