If f (x)=3 x^3+2 x^2 f ^ (1)+x f ^ (2)+ f ^ (3) then f (x)=

Step-by-step solution

Let f(x) = 3x^3 + 2x^2 f'(1) + x f''(2) + f'''(3) and define constants A = f'(1) , B = f''(2) , and C = f'''(3) . The function becomes f(x) = 3x^3 + 2Ax^2 + Bx + C . Successive differentiation yields: f'(x) = 9x^2 + 4Ax + B , f''(x) = 18x + 4A , f'''(x) = 18 . Using the definitions of A , B , and C : A = 9 + 4A + B simplifies to -3A - B = 9 , B = 36 + 4A gives -4A + B = 36 , and C = 18 . Solving the system -3A - B = 9 and -4A + B = 36 : Adding the equations eliminates B , resulting in -7A = 45 , so A = - 45 7 . Sub