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

Options

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

Answer

D

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)`.
(4)
For `xlt0,tin(0,1)i.e.,(1)` should have at least one solution in `tin(0,1)`.
From (1), it is obvious that `ainR^(+)`.
Now, `f_(!)(t)=a(t+1)` represents a straight line. It should meet the curve `f_(1)(t)=t^(2)+3` at least once in `tin(0,1)`.
`f_(1)0=3,f_(1)(1)=4,f_(2)(0)=a,f_(2)(1)=2a`.
If `f_(1)(0)`, then a=3. If `f_(1)(1)=f_(1)(1)`, then a=1.
Hence, `ain(2,3)`.