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What is the smallest distance between the origin and a point on the graph of \(y=\dfrac{1}{\sqrt{2}}\left(x^2-8\right)\)?

 Nov 6, 2022
 #2
avatar+118135 
+1

What is the smallest distance between the origin and a point on the graph below.

I can see that the graph is a concave up parabola'

So I know a smallest distance exists, there will be no maximum distance.

 

\(y=\dfrac{1}{\sqrt{2}}\left(x^2-8\right)\\ \left ( x,\dfrac{1}{\sqrt{2}}\left(x^2-8\right)\right) \text{ is a general point on this graph}\\ \text{I need the distance of this point from (0,0)}\\ d^2=x^2+[\frac{1}{\sqrt{2}}\left(x^2-8\right)]^2\\ d^2=x^2+\frac{1}{2}\left(x^2-8\right)^2\\ \)

 

d will be minimum when d^2 is minimum.  So to avoid confusion I will let  v=d^2

\(v=x^2+\frac{1}{2}\left(x^2-8\right)^2\\\)

\(\frac{dv}{dx }=2x+(x^2-8)*2x\\ \frac{dv}{dx }=2x(1+x^2-8)\\ \frac{dv}{dx }=2x(x^2-7)\\ \frac{dv}{dx }=0\;\;when \;\;x=0\;\;or\;\;x=\pm\sqrt7\\ \)

When x=0   v=32

When x= +/-sqrt7   v=7+0.5=7.5 

Clearly the min is when v=7.5

So the minimum distance is  \(\sqrt{7.5}\)

 

 

LaTex

y=\dfrac{1}{\sqrt{2}}\left(x^2-8\right)\\
\left ( x,\dfrac{1}{\sqrt{2}}\left(x^2-8\right)\right) \text{   is a general point on this graph}\\
\text{I need the distance of this point from (0,0)}\\
d^2=x^2+[\frac{1}{\sqrt{2}}\left(x^2-8\right)]^2\\
d^2=x^2+\frac{1}{2}\left(x^2-8\right)^2\\

 

v=x^2+\frac{1}{2}\left(x^2-8\right)^2\\

 Nov 7, 2022
 #3
avatar+118135 
+1

Checking.

 

The points on the graph  are

   \(x=\pm\sqrt7\\ y=\frac{1}{\sqrt2}(7-8) = \frac{-1}{\sqrt2}=\frac{-\sqrt2}{2}\\ (\sqrt7,\frac{-\sqrt2}{2})\;\;and \;\;(-\sqrt7,\frac{-\sqrt2}{2})\;\;\\ \text{distance from (0,0)}\\ d=\sqrt{7+2/4}=\sqrt{7.5}\)

 

Melody  Nov 7, 2022
 #4
avatar+23 
+1

thank you very much

 Nov 7, 2022

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