If the range of the function f x = s i n - 1 x + 2 t a n - 1 x + x 2 + 4 x + 1 is [ p , q ] , then the value of p + q is

Question

If the range of the function f x = s i n - 1 x + 2 t a n - 1 x + x 2 + 4 x + 1 is [ p , q ] , then the value of p + q is,

MARKS App · Accepted Answer

Domain of f x is - 1 , 1 f x = s i n - 1 x + 2 t a n - 1 x + x 2 + 4 x + 1 f ′ x = 1 1 - x 2 + 2 1 + x 2 + 2 x + 4 = 1 1 - x 2 + 2 1 + x 2 + 2 x + 2 ∀ x ∈ - 1 , 1 , f ′ x > 0 ∴ f x is an increasing function ∴ p = minimum value of f x = s i n - 1 - 1 + 2 t a n - 1 - 1 + - 1 2 + 4 - 1 + 1 = - π 2 + 2 - π 4 - 2 = - π 2 - π 2 - 2 = - π - 2 And q = maximum value of f x = s i n - 1 1 + 2 t a n - 1 1 + 1 + 4 + 1 = - π 2 + 2 π 4 + 6 = π + 6 ∴ So, the range of f x is - π - 2 , π + 6 ∴ p + q = π + 6 - π - 2 = 4

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