+0

# Help

0
88
2

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.

May 17, 2019

#1
+8720
+3

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:

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

May 18, 2019
#2
+10484
+2

Find the semi-major axis.

May 18, 2019