Let f : R R be a function such that f(x)=x^3+x^2 f^ (1)+x f f^ (2)+f^ (3) for x R What is f (1) equal to?

Step-by-step solution

aligned & f(x)=x^3+x^2 f^ (1)+ x f^ (2)+f^ (3)...(1) & f^ (x)=3 x^2+2 x f^ (1)+f^ (2)...(2) & f^ (x)=6 x +2 f^ (1)...(3) & f^ ( x )=6...(4) & f^ (1)=3+2 f^ (1)+f^ (2)...(5) & f^ (2)=12+2 f^ (1)...(6) aligned Using (6) in (5), we get aligned & f^ (1)=3+2 f^ (1)+12+2 f^ (1)-3 f^ (1)=15 & f^ (1)=-5 aligned Using this value in eqn (6) we get aligned & f^ (2)=12+2 (-5) & f^ (2)=2 aligned Using x=3 in eqn (4), f^ (3)=6 Putting value of f^ (1)+f^ (2) and f^ (3) in eqn (1) We get aligned f(x) & =x^3+x^2(-5)+x(2)+6 & =x^3-5