Properties of ITF

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Question

If range of the function $f (x) = \sin^{- 1} x + 2 \tan^{- 1} x + x^{2} + 4 x + 1$ is $[p, q]$ then the value of $p + q$ is

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Solution

$T h e d o m a i n o f f (x) = \sin^{- 1} x + 2 \tan^{- 1} x + x^{2} + 4 x + 1 i s [- 1, 1] . A l s o f (x) i s i n c r e a \sin g i n [- 1, 1] ∴ R a n g e o f f (x) = [f (- 1), f (1)] = [- π - 2, π + 6] = [p, q] ∴ p + q = 4$

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