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Question Number 217660 by ArshadS last updated on 17/Mar/25

f (x) + f (y) = f (x + y) + x y

f (x) = ?

Answered by vnm last updated on 19/Mar/25

f (x) + f (0) = f (x + 0) + x \cdot 0 \Rightarrow f (0) = 0

f (x) + f (- x) = f (x - x) + x (- x) = - x^{2}

f (x) = - \frac{x^{2}}{2} + g (x), g (- x) = - g (x)

g (x) i s a n o d d f u n c t i o n

f (x) - f (x + a) = a x - f (a)

(- \frac{x^{2}}{2} + g (x)) - (- \frac{{(x + a)}^{2}}{2} + g (x + a)) = a x - f (a)

g (x) - g (x + a) = - \frac{a^{2}}{2} - f (a) \forall a, x

g (x) = k x

g (x) i s a d i r e c t p r o p o r t i o n a l i t y

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

Commented by ArshadS last updated on 19/Mar/25

n i c e s i r!

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