Find the minimum value of 9^x - 3^x + 1 over all real numbers x.
The foci of a certain ellipse are at \( (3,10 + \sqrt{105})\) and \((3,10 - \sqrt{105})\). The endpoints of one of the axes are (-5,10) and (11,10).Find the semi-major axis.
+9466 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 = 3x 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:
