Find all values of the parameter a for which the inequality `a.9^x +4(a-1)3^x+a gt 1` is satisfied for all real values of x

Explanation

Putting `t=3^(x)` in the original equation then we obtain

`at^(2)+4(a-1)t+agt1`
`impliesat^(2)+4(a-1)t+(a-1)gt0[tgt0,:'3^(x)gt0]`
This is possible in two cases. First the parabola
`f(t)=at^(2)+4(a-1)t+(a-1)` opens upwards with its vertex(turning point) lying in the non-positive part of the T-axis, as shwon in the following four figures.
`:.agt0` and sum of roots `le0`
`implies-(4(a-1))/(2a)le0` and `f(0)ge0`
`:.agt0,a-1ge0` and `a-1ge0`
Hence `age1`

Second the parabola `f(t)` opens upward with its vertex lying in positive direction of `t` then
`agt0,(4(a-1))/(2a)gt0` and `Dle0`
`impliesagt0,(a-1)lt0`
and `16(a-1)^(2)-4(a-1)ale0`
`impliesagt0,alt1`
an `4(a-1)(3a-4)le0`
`impliesagt0, alt1` and `1leale4/3`
These inequalities cannot have simultaneously.
Hence `age1` from eq. i