Let f: R R be a function defined by f(x)=(2+3 a) x^2+ ( a+2 a-1 ) x+b, a 1 . If f(x+ y )=f(x)+f( y )+1- 2 7 x y , then the value of 28 _ i=1 ^5|f(i)| is

Step-by-step solution

aligned & Put y=0 & f(x)=f(0)+f(x)+1-0 & f(0)=-1 & f(0)=0+0+b & b=-1 & f(-1+1)=f(-1)+f(1)+1+ 2 7 & f(0)=f(-1)+f(1)+ 9 7 aligned aligned -1=(2+3 a)+ ( a+2 a-1 )(-1)+b+(2+3 a) & & + a+2 a-1 +b+ 9 7 aligned aligned & -1=4+6 a-2+ 9 7 & -1=2+ 9 7 +6 a & 6 a=-1-2- 9 7 & a= -5 7 & f(x)= -x^2 7 + 9 7 -12 7 x-1 & f(x)= -x^2 7 - 3 4 x-1 & _ i=1 ^5 f(i)=- 1 7 ( 5 6 11 6 )- 3 4 ( 5 6 2 )-5 & = -55 7 - 45 4 -5 & = 675 28 & 28 | _ i-1 ^5 f(i) |=675 aligned