If f(x) = x ^ 3 - x ^ 2 f ^ ' (1) + x f ^ " (2) - f ^ ' ' ' (3) , x...

Let

$f\left(x\right) = {x}^{3} - a{x}^{2} + bx - c$

${f}^{'}\left(x\right) = 3{x}^{2} - 2ax + b$

${f}^{"}\left(x\right) = 6x - 2a$

{f}^{' ' '}\left(x\right) = 6

Also,

${f}^{'}\left(1\right) = a$

$⇒ 3 - 2a + b = a$

$⇒ 3a = b + 3$

Also,

${f}^{"}\left(2\right) = 12 - 2a = b$

$⇒ 12 - 2a = 3a - 3$

$⇒ 5a = 15 ⇒ a = 3$

$⇒ b = 6$

And,

{f}^{' ' '}\left(3\right) = c ⇒ c = 6

Hence,

$f\left(x\right) = {x}^{3} - 3{x}^{2} + 6x - 6$

Now,

$f\left(3\right) + 2f\left(0\right) = 27 - 27 + 18 - 6 - 12 = 0$

$f\left(2\right) + f\left(1\right) = 8 - 12 + 12 - 6 + 1 - 3 + 6 - 6 = 0$

So,

$f\left(3\right) + 2f\left(0\right) = f\left(2\right) + f\left(1\right)$

$⇒$$2f\left(0\right) - f\left(1\right) + f\left(3\right) = f\left(2\right)$