If a function f : R → R is defined as f x = ∫ x 8 + 4 x 4 - 2 x 2 + 2 d x and f 0 = 1 , then which of the following is correct?

Step-by-step solution

f x = ∫ x 8 + 4 + 4 x 4 - 4 x 4 x 4 - 2 x 2 + 2 d x = ∫ x 4 + 2 2 - 2 x 2 2 x 4 - 2 x 2 + 2 d x = ∫ x 4 + 2 x 2 + 2 x 4 - 2 x 2 + 2 x 4 - 2 x 2 + 2 d x f x = x 5 5 + 2 x 3 3 + 2 x + C f 0 = 0 + 0 + 0 + C = 1 ⇒ C = 1 f x = x 5 5 + 2 x 3 3 + 2 x + 1 Range of f x is R , so f ( x ) is an onto function f - x ≠ f x , so not even f - x ≠ - f x , so not odd f ' x > 0 ∀ x ∈ R , so f ( x ) is one-one