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What is the smallest distance between the origin and a point on the graph of \(y=\dfrac{1}{\sqrt{2}}\left(x^2-3\right)\)?

 

y=(1/√2)*(x^2-3)

 Apr 28, 2019
 #1
avatar+106539 
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y = (1/√2)*(x^2-3)

 

Let  ( a, (1/√2)*(a^2-3) )   be the point we are looking for

 

If the distance is minimized then so is the distance^2.......

 

So

 

D^2  =  (a - 0)^2  + ( (1/√2)*(a^2-3) - 0 )^2   =  a^2  + (1/2)(a^2 - 3)^2

 

Take the dervative wth respect to a and set to 0

 

2a + (a^2 - 3)(2a)  = 0

 

(2a) [ 1 + a^2 - 3] = 0         the first factor set to 0  and solved for a is trivial

 

Setting the second factor to 0  and solving for a produces

 

a^2 - 2  = 0

 

a^2  =  2

 

a =  ±√2

 

So...... (1/√2)*(√2^2 - 3)  =   (1/√2)(-1)  =  -1/√2

 

So....the smallest  distance from either of these points to the origin  =

 

 

√ [ (√2)^2  + (1/√2)^2 ]   =  √  [ 2 + 1/2 ]  =   √5 /√2  =  √10/2   units

 

 

 

cool cool cool

 Apr 28, 2019
edited by CPhill  May 3, 2019

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