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Let f: R R satisfy the equation (x-y) f(x+y)-(x+y) f(x-y)=4 x y (x^2-y^2 ) , y R . If f(1)=-2 , find the absolute difference between maximum and minimum value of f(x) on the interval [- 3 , 3 ] .

Correct answer

4

Step-by-step solution

Put x-y=1 aligned (1) f(2 x-1)-(2 x-1) f(1) & =4(x)(x-1)(2 x-1) f(2 x-1)+2(2 x-1) & =4 x(x-1)(2 x-1) f(2 x-1) & =(2 x-1) (4 x^2-4 x-2 ) f(t) & =t (t^2-3 )=t^3-3 t f^ (t) & =3 t^2-3=0 f(1) & =-2 . f( 3 ) & =0=f(- 3 ) f(-1) & =2 max. aligned |2-(-2)|=4

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