Find all the values of the parameter 'a' for which the inequality `4^x-a2^x-a+3 <= 0` is satisfied by at least one real x.

Explanation

Putting `2^(x)=tgt0` then the original equation reduced in the form
`t^(2)-at-a+3=0`………i
that the quadratic eq. I should have atleast one positive root `(tgt0)` then
Discriminant `D=(-a)^(2)-4.1.(-a+3)ge0`
`impliesa^(2)+4a-12ge0`
`implies(a+6)(a-2)ge0`

`:.a epsilon (-oo,-6]uu[2,oo)`
If roots of eq i are `t_(1)` and `t_(2)` then
`{(t_(1)+t_(2)=a),(t_(1)t_(2)=3-a):}`
For `a epsilon (-oo,-6]`
`t_(1)+t_(2)lt0` and `t_(1)t_(2)gt0`. Therefore both roots are negatie and consequently the original equation has no solutions.
for `a epsilon [2,oo)`
`t_(1)+t_(2)gt0` and `t_(1)t_(2)gt0` consequently atleast one of the roots `t_(1)` or `t_(2)` is greater than zero.
Thus, for `a epsilon [2,oo)` the given equation has atleast one solution.