If a derivable function f: R + arrow R satisfies f ( xy )= f ( x )+ f ( y ) ∀ x , y ∈ R +. If f (16)=3 then (1/ f (2))+(1/ f (4)) is equal to

Question

If a derivable function f: R + arrow R satisfies f ( xy )= f ( x )+ f ( y ) ∀ x , y ∈ R +. If f (16)=3 then (1/ f (2))+(1/ f (4)) is equal to (A) 1 (B) 2 (C) 3 (D) 4

Tardigrade Admin · Accepted Answer

Θ f(x y)=f(x)+f(y) ⇒ f (16)= f (4)+ f (4)=3 ⇒ f (4)=(3/2) ⇒ f (2)+ f (2)=(3/2) ⇒ f (2)=(3/4) ∴ (1/ f (2))+(1/ f (4))=(4/3)+(2/3)=2

Q. If a derivable function $f : R^{+} \to R$ satisfies $f (x y) = f (x) + f (y) \forall x, y \in R^{+}$ . If $f (16) = 3$ then $\frac{1}{f ( 2 )} + \frac{1}{f ( 4 )}$ is equal to

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$Θ f (x y) = f (x) + f (y)$
$\Rightarrow f (16) = f (4) + f (4) = 3 \Rightarrow f (4) = \frac{3}{2}$
$\Rightarrow f (2) + f (2) = \frac{3}{2} \Rightarrow f (2) = \frac{3}{4}$
$∴ \frac{1}{f ( 2 )} + \frac{1}{f ( 4 )} = \frac{4}{3} + \frac{2}{3} = 2$

Q. If a derivable function f:R+→R satisfies f(xy)=f(x)+f(y)∀x,y∈R+. If f(16)=3 then f(2)1​+f(4)1​ is equal to

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Θf(xy)=f(x)+f(y) ⇒f(16)=f(4)+f(4)=3⇒f(4)=23​ ⇒f(2)+f(2)=23​⇒f(2)=43​ ∴f(2)1​+f(4)1​=34​+32​=2

Q. If a derivable function $f : R^{+} \to R$ satisfies $f (x y) = f (x) + f (y) \forall x, y \in R^{+}$ . If $f (16) = 3$ then $\frac{1}{f ( 2 )} + \frac{1}{f ( 4 )}$ is equal to

$Θ f (x y) = f (x) + f (y)$
$\Rightarrow f (16) = f (4) + f (4) = 3 \Rightarrow f (4) = \frac{3}{2}$
$\Rightarrow f (2) + f (2) = \frac{3}{2} \Rightarrow f (2) = \frac{3}{4}$
$∴ \frac{1}{f ( 2 )} + \frac{1}{f ( 4 )} = \frac{4}{3} + \frac{2}{3} = 2$