If the graph of the function f x = a x 3 + x 2 + b x + c is symmetric about the line x = 2 , then the value of a + b is equal to

Step-by-step solution

f x is symmetric about the line x = 2 ∴ f 2 + x = f 2 - x a 2 + x 3 + 2 + x 2 + b 2 + x + c = a 2 - x 3 + 2 - x 2 + b 2 - x + c a 2 + x 3 - 2 - x 3 + 2 + x 2 - 2 - x 2 + b 2 + x - 2 - x = 0 a 8 + 12 x + 6 x 2 + x 3 - 8 - 12 x + 6 x 2 - x 3 + 2 4 x + b 2 x = 0 a 24 x + 2 x 3 + 8 x + 2 b x = 0 2 a x 3 + 24 a + 2 b + 8 x = 0 Which must be true ∀ x ∈ R ∴ it is an identity ∴ 2 a = 0 , 24 a + 2 b + 8 = 0 a = 0 ,   2 b + 8 = 0 b = - 4