If `9^(x+1) + (a^(2)-4a-2) 3^(x) + 1 lt 0 "for all" x in R,` then

Options

A. `a in R`
B. `a in R^(+)`
C. `a in [1, oo)`
D. `a in R -(2)`

Answer

D

Explanation

We have,
`9^(x+1) + (a^(2)-4a-2) 3^(x) + 1 lt 0 "for all" x in R,`
`rArr 9y^(2) + (a^(2)-4a-2) y+1 gt 0 "for all " y gt 0, "where" y = 3^(x)`
`rArr y(9y + (1)/(y) + a^(2) -4a-2) gt 0 "for all" y gt 0`
`rArr (3sqrt(y)-(1)/(sqrt(y)))^(2) + (a-2)^(2) gt 0 "for all" y gt 0`
`rArr a ne 2 rArr a in R -(2)`