Let f(x) is a quadratic function such that f(0)=1 and f(x) d x x^2(x+1)^3 is a rational function, then the value of f^ (0) is ______

Step-by-step solution

aligned & Let g(x)= f(x) d x x^2(x+1)^3 (1) & = ( A x + B x^2 + C x+1 + D (x+1)^2 + E (x+1)^3 ) d x & =A x- B x +C (1+x)- D 1+x - E 2(x+1)^2 aligned Since g(x) is a rational function hence A=C=0g(x)= ( B x^2 + D (x+1)^2 + E (x+1)^3 ) d x (2) Comparing (1) and (2), we get f(x)=(B+D) x^3+(3 B+D+E) x^2+3 B x+B f ( x ) is quadratic aligned & B+D=0, f(0)=1 gives B=1 D=-1 & f(x)=(2+E) x^2+3 x+1 & f^ (0)=3 aligned