A function $f: \mathbf{R} \rightarrow \mathbf{R}$ is such that $f(\mathrm{l})=2$ and $f(x+y)=f(x) \cdot f(y) \forall x, y$. The area (in square units) enclosed by the lines $2|x|+5|y| \leq 4$ expressed interms of $f(1), f(2)$ and $f(4)$ is

• A

  $\frac{f(4)}{f(1)+2 f(2)}$
• Correct answer
  B

  $\frac{f(4)}{1+f(2)}$
• C

  $\frac{2 f(4)}{2 f(1)+f(2)}$
• D

  $\frac{f(4)}{2 f(1)+f(2)}$

Solution

Given, $f(x+y)=f(x) \cdot f(y)$ $\therefore \quad f(x)=a^x$ $f(\mathrm{1})=a^{\mathrm{l}}=2 \Rightarrow a=2$ $\therefore \quad f(x)=2^x$ Area enclosed by the lines $2|x|+5|y| \leq 4$ $4\left(\frac{1}{2} \times 2 \times \frac{4}{5}\right)=\frac{16}{5}$
$=\frac{16}{1+(2)^2}=\frac{(2)^4}{1+(2)^2}=\frac{f(4)}{1+f(2)}$

Hence, the correct answer is option: 'B'.