If f x = x 3 2 x 2 + 1 1 + 3 x 3 x 2 + 2 2 x x 3 + 6 x 3 − x 4 x 2 − 2 for all x ∈ ℝ , then 2 f 0 + f ' 0 is equal to

Step-by-step solution

Given: f x = x 3 2 x 2 + 1 1 + 3 x 3 x 2 + 2 2 x x 3 + 6 x 3 - x 4 x 2 - 2 ⇒ f 0 = 0 1 1 2 0 6 0 4 - 2 ⇒ f 0 = 0 + 4 + 8 ⇒ f 0 = 12 ⇒ f ' x = x 3 2 x 2 + 1 1 + 3 x 3 x 2 + 2 2 x x 3 + 6 3 x 2 - 1 0 2 x + x 3 2 x 2 + 1 1 + 3 x 6 x 2 3 x 2 x 3 - x 4 x 2 - 2 + 3 x 2 4 x 3 3 x 2 + 2 2 x x 3 + 6 x 3 - x 4 x 2 - 2 ⇒ f ' 0 = 0 1 1 2 0 6 - 1 0 0 + 0 1 1 0 2 0 0 4 - 2 + 0 0 3 2 0 6 0 4 - 2 ⇒ f ' 0 = 0 - 0 + 6 + 0 + 0 + 0 + 0 + 3 8 ⇒ f ' 0 = - 6 + 24 = 18 ⇒ 2 f 0 + f ' 0 = 42