Let $f : R^{+} \to R^{+}$ be a differentiable function satisfying $f (x y) = \frac{f (x)}{y} + \frac{f (y)}{x}$ for all $x, y ϵ R^{+}$ . Also $f (1) = 0, f^{'} (1) = 1$ . If M is the greatest value of $f (x)$ then $[m + e]$ is ___ (where [.] represents Greatest Integer Function).

Join BYJU'S Learning Program

Let f be a function defined on the set of all positive integers such that f(xy)=f(x)+f(y) for all positive integers x,y. If f(12)=24 and f(8)=15, the value of f(48) is

The correct option is D 34 Let x=2,y=2f(4)=f(2)+f(2)⇒f(4)=2f(2)⇒f(2×4)=f(2)+f(4)=15⇒3f(2)=15⇒f(2)=5Now, f(48)=f(12×4)=f(12)+f(4)=24+2×5=34

Let $f$ be a function defined on the set of all positive integers such that $f (x y) = f (x) + f (y)$ for all positive integers $x, y$ . If $f (12) = 24$ and $f (8) = 15$ , the value of $f (48)$ is

The correct option is D $34$
$Let x = 2, y = 2 f (4) = f (2) + f (2) \Rightarrow f (4) = 2 f (2) \Rightarrow f (2 \times 4) = f (2) + f (4) = 15 \Rightarrow 3 f (2) = 15 \Rightarrow f (2) = 5 Now, f (48) = f (12 \times 4) = f (12) + f (4) = 24 + 2 \times 5 = 34$