Find-all-functions-f-R-R-such-that-f-x-y-2f-x-3f-y-4xyf-2x-3y-x-y-R-

Question Number 142570 by mathsuji last updated on 02/Jun/21

F i n d a l l f u n c t i o n s f : R \to R s u c h t h a t

f (x + y) = 2 f (x) + 3 f (y) - 4 x y f (2 x - 3 y)

(\forall x; y \in R)

Answered by ajfour last updated on 02/Jun/21

f (x) = 5 f (\frac{x}{2}) - x^{2} f (- \frac{x}{2})

a l s o

f (x) = 2 f (x) + 3 f (0)

a n d

f (0) = 5 f (0) \Rightarrow f (0) = 0

\Rightarrow f (x) = 0

a n d

f (2 x) = 5 f (x) - 4 x^{2} f (- x)

f (- 2 x) = 5 f (- x) - 4 x^{2} f (x)

s a y f (2 x) = A, f (- 2 x) = B

f (x) = \frac{5 A + 4 x^{2} B}{25 - 16 x^{4}}

f (- x) = \frac{4 x^{2} A + 5 B}{25 - 16 x^{4}}

\Rightarrow f (x) = f (- x)

\Rightarrow f (2 x) = (5 - 4 x^{2}) f (x)

\dots \dots . .

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