Considering only the principal values of inverse trigonometric functions, the number of positive real values of x satisfying tan - 1 ( x ) + tan - 1 ( 2 x ) = π 4 is :

Step-by-step solution

Let, f x = tan - 1 x + tan - 1 2 x Now, differentiating the above function we get, ⇒ f ' x = 1 1 + x 2 + 2 1 + 4 x 2 ⇒ f ' x > 0 ; ∀ x ∈ R ⇒ lim x → ∞ f x = π and lim x → - ∞ f x = - π Hence, f x is monotonically increasing for x ∈ R and the range of f x is - π , π . Hence, f x = π 4 has exactly one solution for x > 0 .