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Let f : [0, 1]toR be such that f(xy)=f(x...

Let `f : [0, 1]toR` be such that `f(xy)=f(x)xxf(y)` for all x, `y in [0,1]" and " f(0)ne0." if ' y=y(x)` satisfues the differential equation, `dy/dx=f(x)" with"y(0)=1," then "y(1/4)+y(3/4)` is

A

5

B

4

C

2

D

3

Text Solution

Verified by Experts

The correct Answer is:
D
Since `r(0)ne0" "(f(0))^(2)=f(0)`
`rArr" "f(0)=1`
Put`" "y=0`
`f(x).f(0)=f(0)`
`rArr" "f(x).f(0)=f(0)`
`rArr" "f(x)=1 AA x`
`(dy)/(dx)=1" "rArr" "y=x+c`
`y(0)=1" "rArr" "c=1`
`y=x+1`
`rArr" "y((1)/(9))+y((3)/(4))=3`
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