Consider the inequation `9^(x) -a3^(x) - a+ 3 le 0`, where a is real parameter.
The given inequality has at least one negative solution for `a in ` (a) `(-oo,2)` (b) `(3,oo)` (c) `(-2,oo)` (d) `(2,3)`

Options

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

Answer

B

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)`.
(2)
In this case, both graphs should meet at least once in `tin(0,oo)`.
For a=2, both curves touch. Hence, `ain[2,oo)`.