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

Given that `9^(x) - a3^(x) - a+3 le 0`
Let `t = 3^(x)`. Then,
`t^(2) -at - a + 3 le 0`
or `t^(3) + 3 le a(t+1) " "(1)`
where `t in R^(+), AA x in R`
Let `f_(1)(t) = t^(2) + 3` and
`f_(2)(t) = a(t+1)`.
For at least one posititve solution, `t in (1,oo)`. That means graphs of `f_(1)(t) = t^(2) + 3 and f_(2) (t) = a(t+1)` should meet at least once is `t in (1,oo)`. If `a = 2`, both the curves touch each other at `(1,4)`.
Hence, the required range is `a in (2,oo)`