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Find the range of `f(x) = (sin^(-1) x)^(2) + 2pi cos^(-1) x + pi^(2)` |
Answer» `f(x) = (sin^(-1) x)^(2) + 2pi ((pi)/(2) - sin^(-1) x) + pi^(2)` `= (sin^(-1) x)^(2) + pi^(2) - 2pi sin^(-1) x + pi^(2)` `= (sin^(-1) x - pi)^(2) + pi^(2)` `f_("min") = ((pi)/(2) - pi)^(2) + pi^(2) = (pi^(2))/(4) + pi^(2) = (5 pi^(2))/(4)` `f_("max") = (-(pi)/(2) -pi)^(2) + pi^(2) = (9pi^(2))/(4) + pi^(2) = (13 pi^(2))/(4)` Therefore, range is `[(5 pi^(2))/(4), (13 pi^(2))/(4)]` |
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