Consider the inequation `9^(x) -a3^(x) - a+ 3 le 0`, where a is real parameter.
The given inequality has at least one real solutions for `a in `.

Options

A. `(-oo,-2)`
B. `(3,oo)`
C. `(2,oo)`
D. `[-2,oo)`

Answer

C

Explanation

`9^(x)-a*3-a+le0`
Let `t=3^(x)`. Then
`t^(2)-at-a+3le0`
`ort^(2)+3lea(t+1)` (1)
where `tinR^(+)" for "AAx inR`,

`"Let"f_(1)(t)" be "t^(2)+3andf_(2)(t)" be "a(t+1)`.
(3)
For at least one positive solution, `tin(1,oo)`. Therefore meet at least graphs of `f_(1)(t)=t^(2)+3andf_(2)(t)=a(t+1)` should meet at least once in `tin(1,oo)`.
If a=2, both the curves touch each other at (1,4).
Hence, `ain(2,oo)`.