For `x, y, z, t in R, sin^(-1) x + cos^(-1) y + sec^(-1) z ge t^(2) - sqrt(2pi t) + 3pi`
The principal value of `cos^(-1) (cos 5t^(2))` is

Options

A. `(3pi)/(2)`
B. `(pi)/(2)`
C. `(pi)/(3)`
D. `(2pi)/(3)`

Answer

B

Explanation

`sin^(-1) x in [-(pi)/(2), (pi)/(2)]`
`cos^(-1) y in [0, pi]`
`sec^(-1) z in [0, (pi)/(2)) uu ((pi)/(2), pi]`
`rArr sin^(-1) x + cos^(-1) y + sec^(-1) z le (pi)/(2) + pi + pi = (5pi)/(2)`
Also `t^(2) - sqrt(2pi) t + 3pi = t^(2) -2 sqrt((pi)/(2)) t + (pi)/(2) - (pi)/(2) + 3pi`
`= (t -sqrt((pi)/(2)))^(2) + (5pi)/(2) ge (5pi)/(2)`
The given inequation exists if equality holds, i.e.,
L.H.S. = R.H.S. `= (5pi)/(2)`
`rArr x = 1, y = -1, z = -1 and t = sqrt((pi)/(2))`
`rArr cos^(-1) (cos 5t^(2)) = cos^(-1) (cos 5 t^(2)) = cos^(-1) (cos ((5pi)/(2))) = (pi)/(2)`
`cos^(-1) ("min") {x, y, z}) = cos^(-1) (-1) = pi`