for-x-y-R-given-f-x-f-2x-y-5xy-f-3x-y-x-2-1-find-f-10-

Question Number 80861 by jagoll last updated on 07/Feb/20

f o r x, y \in R

g i v e n f (x) + f (2 x + y) + 5 x y =

f (3 x - y) + x^{2} + 1

f i n d f (10)

Commented by jagoll last updated on 07/Feb/20

i w o r k b y

p u t y = 0

f (x) + f (2 x) = f (3 x) + x^{2} + 1

l e t 3 x = t

f (\frac{t}{3}) + f (\frac{t}{2}) = f (t) + \frac{t^{2}}{9} + 1

Commented by jagoll last updated on 07/Feb/20

m y w a y i s r i g h t ?

Commented by ~blr237~ last updated on 07/Feb/20

y e s

Commented by mr W last updated on 07/Feb/20

f (x) + f (2 x + y) + 5 x y = f (3 x - y) + x^{2} + 1

y = 0 :

f (x) + f (2 x) = f (3 x) + x^{2} + 1 \dots (i)

y = x :

f (x) + f (3 x) + 5 x^{2} = f (2 x) + x^{2} + 1 \dots (i i)

(i) + (i i) :

2 f (x) + 5 x^{2} = 2 (x^{2} + 1)

\Rightarrow f (x) = 1 - \frac{3 x^{2}}{2}

\Rightarrow f (10) = 1 - \frac{3 \times 10^{2}}{2} = - 149

Commented by jagoll last updated on 07/Feb/20

t h a n k y o u m r w a n d m r b l r

Answered by ~blr237~ last updated on 07/Feb/20

t a k i n g y = 0 w e h a v e f (x) + f (2 x) = f (3 x) + x^{2} + 1 (1)

t a k i n g x = y w e h a v e f (x) + f (3 x) + 5 x^{2} = f (2 x) + x^{2} + 1 (2)

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

f (x) = - \frac{3}{2} x^{2} + 1

f (10) = - \frac{3}{2} \times 100 + 1 = - 149

Facebook Tweet Pin