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Question

If a function satisfies (xy)f(x+y)(x+y)f(xy)=2(x2yy3),x,yR and f(1)=2, then

A
f(x) must be polynomial function
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B
f(3)=12
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C
f(0)=0
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D
f(x) may not be differentiable
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Solution

The correct options are
A f(x) must be polynomial function
B f(3)=12
C f(0)=0
(xy)f(x+y)(x+y)f(xy) =2y((xy)(x+y))

Let xy=u,x+y=v

uf(v)vf(u)=uv(vu)

f(v)vf(u)u=vu

(f(v)vv)=(f(u)uu)=constant

Let f(x)xx=λ

f(x)=(λx+x2)

f(1)=2

λ+1=2λ=1

f(x)=x2+x

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