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Let f:R in R be a function given by f(x+...

Let `f:R in R` be a function given by `f(x+y)=f(x) f(y)"for all"x,y in R`
`"If "f(x) ne 0,"for all "x in R and f'(0)=log 2,"then "f(x)=`

A

`x^(2)`

B

`2^(x)`

C

`x(log 2)`

D

`e^(2x)`

Text Solution

Verified by Experts

The correct Answer is:
B
Proceeding as in Example 33, we have
`f'(x)=f(x). log_(e)2`
`Rightarrow (f'(x))/(f(x))=log_(e)2`
`Rightarrow log_(e)f(x)=x log_(e)2+log_(e)C " "["On integrating"]`
`Rightarrow f(x)=C 2^(x)
`"But "f(0)=1"Therefore "C=1`
"Hence", `f(x)=2^(x)`
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