Let, f: R → R be a function such that f(x) = x ^ 3 + x ^ 2 f ' (1)...

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

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

$⇒ f '' \left(x\right) = 6x + 2a$

$⇒ f ''' \left(x\right) = 6$

According to the question,

$a = f ' \left(1\right) = 3 + 2a + b ⇒ a + b = - 3 . . . \left(i\right)$

$b = f '' \left(2\right) = 12 + 2a ⇒ 2a - b = - 12 . . . \left(ii\right)$

$c = f ''' \left(3\right) ⇒ c = 6$

Solving equations \left(i\right) \text{ and } \left(ii\right)$, we get,$a = - 5 \text{ and } b = 2$</p><p>$⇒ f\left(x\right) = {x}^{3} - 5{x}^{2} + 2x + 6$</p><p>$∴ f\left(2\right) = 8 - 20 + 4 + 6 = - 2 .$