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If f : R → R be a function such that f x = x 3 + x 2 + 4 x + sin ⁡ x . Then, the function f x is

Options

  1. Aone-one and onto
  2. Bone-one and into
  3. Cmany-one and onto
  4. Dmany-one and into

Correct answer

A. one-one and onto

Step-by-step solution

f x = x 3 + x 2 + 4 x + sin ⁡ x ⇒ f ' x = 3 x 2 + 2 x + 4 + cos ⁡ x f ' x = 3 x + 1 3 2 + 11 9 - - cos ⁡ x > 0 as   3   x + 1 3 2 + 11 9 min = 11 3 and - cos ⁡ x has the maximum value 1 . ⇒ f x is strictly increasing and hence it is one-one Also, l i m x → ∞ f x ⇒ ∞ and l i m x → - ∞ f x ⇒ - ∞ . Thus, the range of f x is R , hence it is onto.

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