If `x , y , z in [-1,1]` such that `sin^(-1)x+sin^(-1)y+sin^(-1)z=-(3pi)/2,` find the value of `x^2+y^2+z^2dot`

Options

A. 1
B. 3
C. `(3pi^(23))/(4)`
D. `3pi^(2)`

Answer

B

Explanation

We know that the minimum value of `sin^(-1) x for x in[-1,1]` is `-(pi)/(2)`
`therefore sin^(-1)xge-(pi)/(2)sin^(-1)yge-(pi)/(2) and sin^(-1)z ge -(pi)/(2)` for all
`rarr sin^(-1)x+sin^(-1)y+sin^(-1)zge(pi)/(2)+(pi)/(2)+(pi)/(2)`
` therefore sin^(-1)x+sin^(-1)y+sin^(-1)z=(3pi)/(2)`
`rarr sin^(-1)x=(pi)/(2)sin^(-1)y -(pi)/(2),sin^(-1)z=(pi)/(2)`
`rarr x==z=-1`
Hence `x^(2)+y^(2)+z^(2)=(1)^(2)+(-1)^(2)+(-1)^(2)=3`