Determine all function f:R→R which satisfy f(a+x)−f(a−x)=4ax for all real a and x.


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Question Number 120300 by bemath last updated on 30/Oct/20

$D e t e r m i n e a l l f u n c t i o n f : R \to R$ $w h i c h s a t i s f y f (a + x) - f (a - x) = 4 a x$ $f o r a l l r e a l a a n d x .$
Commented by benjo_mathlover last updated on 30/Oct/20

$l e t f (x) = {p x}^{2} + q x + r$ $f (a + x) = p (a^{2} + 2 a x + x^{2}) + q x + a q + r$ $f (a + x) = {p x}^{2} + (2 a p + q) x + a^{2} p + a q + r$ $f (a - x) = p (a^{2} - 2 a x + x^{2}) + a q - q x + r$ $f (a - x) = {p x}^{2} - (2 a p + q) x + a^{2} p + a q + r$ $f (a + x) - f (a - x) = (4 a p + 2 q) x \equiv 4 a x$ $\to 4 a p + 2 q = 4 a; 4 a p - 4 a = - 2 q$ $4 a (1 - p) = 2 q \Rightarrow 1 - p = \frac{q}{2 a}$ $\Rightarrow p = 1 - \frac{q}{2 a} = \frac{2 a - q}{2 a}$ $f (x) = (\frac{2 a - q}{2 a}) x^{2} + q x + r$
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