If α is an integer satisfying ∣ α ∣ ≤ 5 − ∣ [ x ] ∣ , where x is a real number for which 2 x tan − 1 x is greater than or equal to ln ( 1 + x 2 ) , then the number of maximum non-negative possible val

Question

If α is an integer satisfying ∣ α ∣ ≤ 5 − ∣ [ x ] ∣ , where x is a real number for which 2 x tan − 1 x is greater than or equal to ln ( 1 + x 2 ) , then the number of maximum non-negative possible values of α is [0pt] (where [.] denotes the greater integer function)

TestHub · Accepted Answer

Let y = 2 x tan − 1 x − ln ( 1 + x 2 ) y ′ = 2 tan − 1 x + 1 + x 2 2 x ​ − 1 + x 2 2 x ​ ⇒ y ′ > 0∀ x ∈ R + , y ′ < 0∀ x ∈ R − ⇒ y ≥ 0 , ∀ x ∈ R ∴ 5 − ∣ [ x ] ∣ takes the values 0 , 1 , 2 , 3 , 4 , 5. { ∵ ∣ α ∣ ≤ 5 − ∣ [ x ] ∣ } ∴ ∣ α ∣ ≤ 5 − ∣ [ x ] ∣ is satisfied by α = 0 , ± 1 , ± 2 , ± 3 , ± 4 , ± 5

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