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 (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 ​ This implies that y ′ > 0 for all x ∈ R + and y ′ < 0 for all x ∈ R − . Therefore, y ≥ 0 for all x ∈ R . Thus, 5 − ∣ [ x ] ∣ takes the values 0 , 1 , 2 , 3 , 4 , 5 , because ∣ α ∣ ≤ 5 − ∣ [ x ] ∣ . Therefore, ∣ α ∣ ≤ 5 − ∣ [ x ] ∣ is satisfied by α = 0 , ± 1 , ± 2 , ± 3 , ± 4 , ± 5 .

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