If `sin^(-1)x_i in [0,1]AAi=1,2,3, .28` then find the maximum value of `sqrt(sin^(-1)x_1)sqrt(cos^(-1)x_2)+sqrt(sin^(-1)x_2)sqrt(cos^(-1)x_3)+` `sqrt(

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If `sin^(-1)x_i in [0,1]AAi=1,2,3, .28` then find the maximum value of `sqrt(sin^(-1)x_1)sqrt(cos^(-1)x_2)+sqrt(sin^(-1)x_2)sqrt(cos^(-1)x_3)+` `sqrt(

If `sin^(-1)x_i in [0,1]AAi=1,2,3, .28` then find the maximum value of `sqrt(sin^(-1)x_1)sqrt(cos^(-1)x_2)+sqrt(sin^(-1)x_2)sqrt(cos^(-1)x_3)+` `sqrt(sin^(-1)x_3)sqrt(cos^(-1)x_4)++sqrt(sin^(-1)x_(28))sqrt(cos^(-1)x_1)`

Monjur Alahi

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2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

`E = sqrt(sin^(-1)x_(1)) sqrt(cos^(-1) x_(2)) + sqrt(sin^(-1) x_(2)) sqrt(cos^(-1) x_(3)) + sqrt(sin^(-1) x_(3)) sqrt(cos^(-1) x_(4)) +...+ sqrt(sin^(-1) x_(28)) sqrt(cos^(-1) x_(1))`
`x_(i) in [0,1] AA i = 1, 2, 3, .., 28`
`:. sin^(-1) x_(i) gt 0`
Now using `A.M. ge G.M`., we have
`(a^(2) + b^(2))/(2) ge ab`, where `a, b gt 0`
`:. sqrt(sin^(-1) x_(1)) sqrt(cos^(-1) x_(2)) le ((sin^(-1) x_(1) + cos^(-1) x_(2))/(2))`
`{:(sqrt(sin^(-1) x_(2)) sqrt(cos^(-1) x_(3)) lt ((sin^(-1) x_(2) + cos^(-1) x_(3))/(2))),(vdots" "vdots),(sqrt(sin^(-1) x_(28)) sqrt(cos^(-1) x_(1)) le ((sin^(-1) x_(28) + cos^(-1) x_(1))/(2))):}`
On adding all, we get
`E le underset(i =1)overset(28)sum (sin^(-1) x_(i) + cos^(-1) x_(i))/(2)`
`:. E le (28((pi)/(2)))/(2)`
`:. E_("max") = 7 pi`