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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
avatar+8209 
+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
avatar+10409 
+2

Find the semi-major axis.

laugh

 May 18, 2019

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