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Let \mathrm{f(x+y)=f(x) f(y)} for all \mathrm{x} and \mathrm{y}. If \mathrm{f(5)=2} and \mathrm{f^{\prime}(0)=3} then \mathrm{f^{\prime}(5)} is equal to

Option: 1

5


Option: 2

8


Option: 3

0


Option: 4

none of these.


Answers (1)

best_answer

Putting \mathrm{x=0, y=5} in the given equation, we get

\mathrm{f(0+5)=f(0) f(5) \Rightarrow f(5)[f(1)-0]=0 \Rightarrow f(0)=1}

\mathrm{f^{\prime}(5)=\lim _{h \rightarrow 0} \frac{f(5+h)-f(5)}{h}=\lim _{h \rightarrow 0} \frac{f(5)(f(h)-1)}{h}}
\mathrm{=f^{\prime}(5) \lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h}=(2)(3)=6}.

Posted by

Sanket Gandhi

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