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What is the smallest distance between the origin and a point on the graph of  y=1/sqrt(2)  + (x^2-3)?

 Feb 2, 2021
 #1
avatar+11667 
+1

What is the smallest distance between the origin and a point on the graph of  y=1/sqrt(2)  + (x^2-3)?

 

Hello Guest!

 

\(y=\frac{1}{\sqrt{2}}+x^2-3\\ y'=2x\\ y'_{ ↲}=-\frac{1}{2}x\\ [the\ perpendicular\ on\ y '\ to\ the\ origin\ of\ the\ coordinates ] \)

 

\( \color{blue}x^2+\frac{1}{\sqrt{2}}-3=-\frac{1}{2}x\\x^2+ \frac{1}{2}x+\frac{1}{\sqrt{2}}-3=0\)

          p              q

 

\(x=-\frac{p}{2}\pm \sqrt{(\frac{p}{2})^2-q}\\ x=-\frac{1}{4}\pm \sqrt{(\frac{1}{4})^2-\frac{1}{\sqrt{2}}+3)}\)

\(\color{blue}x_1=1.2847\\ x_2=-1.7847\ [not\ applicable ]\)

\(y_1=\frac{1}{\sqrt{2}}+x^2-3=\frac{1}{\sqrt{2}}+1.2847^2-3\)

\(y_1=-0.64236\)

 

The smallest distance

\(=\sqrt{x_1^2+y_1^2}=\sqrt{1.2847^2+(-0.64236)^2}\)

\(=1.43637\)

laugh  !

 Feb 2, 2021
edited by asinus  Feb 2, 2021
edited by asinus  Feb 2, 2021
 #2
avatar+32312 
+3

What is the smallest distance between the origin and a point on the graph of  y=1/sqrt(2)  + (x^2-3)?

 

 Feb 2, 2021

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