Consider the inequality, `9^(x)-a.3^(x)-a+3 le 0`, where `'a'` is a real parameter.
(a) Find the value of `'a'` for which the inequality has at least one negative solution.
(b) Find the values of `'a'` for which the inequality has at least one positive solution.
(c) Find the vlaues of `'a'` for which the inequality has at least one real solution.

Options

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