Question

# Let $$f\left( xy \right) =f\left( x \right) \cdot f\left( y \right)$$ for all $$x,y\in R$$. If $$f^{ ' }\left( 1 \right) =2$$ and $$f\left( 4 \right) =4$$, then $$f^{ ' }\left( 4 \right)$$ equal to

A
4
B
1
C
12
D
8

Solution

## The correct option is D $$8$$We have $$f\left( x,y \right) =f\left( x \right) f\left( y \right)$$ for all $$x,y\in R$$.Putting $$x=y=1$$, we get$$f\left( 1 \right) =f\left( 1 \right) f\left( 1 \right)$$$$\Rightarrow f\left( 1 \right) \left[ 1-f\left( 1 \right) \right] =0$$$$\Rightarrow f\left( 1 \right) =1$$             $$\left[ \because f\left( 1 \right) \neq 0 \right]$$Now, $$f^{ ' }\left( 1 \right) =2$$$$\Rightarrow \displaystyle\lim _{ h\rightarrow 0 }{ \dfrac { f\left( 1+h \right) -f\left( 1 \right) }{ h } } =2$$$$\Rightarrow f\left( 1 \right) \displaystyle\lim _{ h\rightarrow 0 }{ \dfrac { f\left( h \right) -1 }{ h } } =2$$$$\Rightarrow \displaystyle\lim _{ h\rightarrow 0 }{ \dfrac { f\left( 1 \right) f\left( h \right) -f\left( 1 \right) }{ h } } =2$$          [using $$f\left( 1 \right) =1$$]$$\Rightarrow \displaystyle\lim _{ h\rightarrow 0 }{ \dfrac { f\left( h \right) -1 }{ h } } =2$$                    ....(i)Now, $$f^{ ' }\left( 4 \right) =\displaystyle\lim _{ h\rightarrow 0 }{ \dfrac { f\left( 4+h \right) -f\left( 4 \right) }{ h } }$$$$=\displaystyle\lim _{ h\rightarrow 0 }{ \dfrac { f\left( 4 \right) \cdot f\left( h \right) -f\left( 4 \right) }{ h } }$$$$=\left\{ \displaystyle\lim _{ h\rightarrow 0 }{ \dfrac { f\left( h \right) -1 }{ h } } \right\} \cdot f\left( 4 \right) =2f\left( 4 \right)$$      ....{from (i)}$$=2\times 4=8$$Mathematics

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