If [ array lll 1 & 0 & 0 6 & 2 & 0 5 & 4 & 3 array ] [ array c x x ^2 1 array ]= [ array c x 2 ax + bx 5 x + cx ^2+3 array ] x R and f( x ) is a differentiable function satisfying f( x )+f( y )=f ( x + y 1- xy ) for all x , y R ,( xy 1) and _ x 0 f( x ) x =2 , then find the value of [ ₀^1 ax x ^2+ bx + c f(1) dx ] , where [.] denotes greatest integer function.

Step-by-step solution

. array l 6 x +2 x ^2=2 ax + bx ^2 5 x +4 x ^2+3=5 x + cx ^2+3 array a =3, ~b =2 & c =4 Given functional relation is aligned & f(x)+f(y)=f ( x + y 1- xy ) & x = y =0, f(0)=0 & y =- x & f( x )+f(- x )=f(0) & f( x )=-f(- x ) aligned aligned & Now, f^ (x)= _ h 0 f(x+h)-f(x) h & = _ h 0 f(x+h)+f(-x) h & = _ h 0 f ( x + h - x 1-( x + h )(- x ) ) h = _ h 0 f ( ~h 1+ x ( x + h ) ) h & = _ h 0 f ( ~h 1+ x ( x + h ) ) ( h 1+ x ( x + h ) ) ( 1 1+ x ( x + h ) ) & f^ (x)= 2 1+x^2 _ x 0 f(x) x =2 & f( x )=2 ⁻¹ x + c & f(0)=0+c