Question

If f(x + y) = f(x) f(y) for all x and y and f(5) = 2, f'(0) = 3, then f'(5) is

• A. 5
• B. 6
• C. 0
• D. none of these

Answer 1

Answer: (B)

$f\left(x+y\right)=f\left(x\right)f\left(y\right)$
$\Rightarrow f\left(x+5\right)=f\left(x\right)f\left(5\right)=2f\left(x\right)$
$\Rightarrow f^{\prime }\left(x+5\right)=2f^{\prime }\left(x\right)$
$\Rightarrow f^{\prime }\left(0+5\right)=2f^{\prime }\left(0\right)=2\left(3\right)=6\Rightarrow f^{\prime }\left(5\right)=6$