If `cos x - y^(2) - sqrt(y - x ^(2) - 1 )ge 0 ` , then

Options

A. ` y ge 1 `
B. `x in ` R
C. `y = 1 `
D. `x = 0`

Answer

3,4

Explanation

` cos x- y^(2) - sqrt(y - x^(2) - 1 ) ge 0 ` (1)
Now , `sqrt(y - x^(2) - 1)` is defined when ` y - x^(2) - 1 ge 0 or ge x^(2) + 1 `
so minimun values of y is 1. From (1),
` cos x - y^(2) ge sqrt(y - x^(2) - 1)`
where `cos x - y^(2) ge 0 ` [ as when cos x is maximum `(=1) and y^(2) ` is
minimum (=1), so cos x - `Y^(2)` is maximum ] . Also,
` sqrt(y - x ^(2) - 1)`
Hence,
`cos x - y^(2) = sqrt(y - x ^(2) - 1)= 0 `
`rArr y = 1 cos x = 1 , y = x^(2) = 1 `
`rArr x = 0, y = 1 `