Question

Let f(x) be a continuous and differentiable function and f(y)f(x + y) = f(x) for all real values of x and y. If f(5) = 3 and f'(3) = 7 then the value of f'(8) is

• A. $0$
• B. $\frac{1}{7}$
• C. $\frac{7}{3}$
• D. $7$

Answer 1

The correct option is C $\frac{7}{3}$

f(y)f(x + y) = f(x)
Putting y = 5
f(5)f(x + 5) = f(x)
$\Rightarrow 3f(x+5)=f\left(x\right)$
$\Rightarrow 3f^{\prime }(x+5)=f^{\prime }\left(x\right)$
$\therefore f^{\prime }\left(8\right)=\frac{f^{\prime }\left(3\right)}{3}=\frac{7}{3}$