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# graphs

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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
+11086
+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

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

$$=1.43637$$

!

Feb 2, 2021
edited by asinus  Feb 2, 2021
edited by asinus  Feb 2, 2021
#2
+31716
+2

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