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Let   $y\,=\,9^x - 3^x + 1$

$y\,=\,(3^2)^x - 3^x + 1\\~\\ y\,=\,3^{2x} - 3^x + 1\\~\\ y\,=\,{\color{}(3^x)}^2-{\color{}(3^x)}+1$

Notice that this is a quadratic equation. We can let  u = 3^x  to make it clearer.

$y\,=\,u^2-u+1\\~\\ y\,=\,u^2-u+\frac14-\frac14+1\\~\\ y\,=\,(u-\frac12)^2-\frac14+1$      We want to find what value of  u  minimizes  y .

When the quadratic equation is in this form we can see that the minimum value of  y  occurs when...

$u\,=\,\frac12\\~\\ 3^x\,=\,\frac12\\~\\ x\,=\,\log_3(\frac12)$

And when   $x\,=\,\log_3(\frac12)$ ,

  $y\,=\,(3^x)^2-(3^x)+1\,=\,(\frac12)^2-\frac12+1\,=\,\frac34$

By looking at a graph, we can see this is the minimum:

https://www.desmos.com/calculator/s5ikzljydn

Find the semi-major axis.