Let f: R R be a function such that f (x)=x^3+x^2 f ^ (1)+x f ^ (2)+6, x R , then f (2) equals

Step-by-step solution

f (x)=x^3+x^2 f ^ (1)+x f ^ (2)+6 f ^ (x)=3 x^2+2 x f ^ (1)+ f ^ (2) ...(i) f ^ (x)=6 x+2 f ^ (1) ...(ii) Substituting x=1 in (i), we get f^ (1)=3(1)^2+2(1) f^ (1)+f^ (2) f ^ (1)+ f ^ (2)=-3 ...(iii) Substituting x=2 in (ii), we get f^ (2)=6(2)+2 f^ (1) f ^ (2)=12+2 f ^ (1) ...(iv) From (iii) and (iv), we get aligned & f ^ (1)+12+2 f ^ (1)=-3 & 3 f ^ (1)=-15 & f ^ (1)=-5 & From (iii) ,-5+ f ^ (2)=-3 & f ^ (2)=2 aligned aligned f(2) & =2^3+2^2(-5)+2(2)+6 & =8-20+4+6 & =-2 aligned