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

I got sqrt(5/2), but it was wrong...

May 31, 2021

#1
+1

I  get  the  same result    .....

Let  the  point we  are looking for  be    ( x ,        (x^2 - 3) /  sqrt 2   )

We can  use  the  square of  the distance  formula

D^2  =   x^2   +  (x^2 - 3)^2  / 2         simplify

D^2  = x^2  +  (1/2)(x^4 - 6x^2 + 9)

D^2  =  x^2  + (1/2) x^4  -  3x^2  +  9/2

D^2  =   (1/2)x^4  - 2x^2  + 9/2   take  the  derivative of  the  function  and set to  0

2x^3   -  4x   =  0

x^3  -  2x  =  0        factor

x( x^2  - 2)   =   0

Setting  the  second factor to  0   and solving for  x  we  get  x =  sqrt 2   or  x  =  -sqrt 2

Either  value  minimizes the  distance....so.....one  point  is  (sqrt 2 , -1/ sqrt 2)

And  the  distance =   sqrt ( 2  + (1/2)   =   sqrt  ( 2.5)   =  sqrt (5/2)    ≈   1.58

Here's  a graph :  https://www.desmos.com/calculator/3zo8v8cxyz   May 31, 2021
#2
+1

The answer was sqrt10/2 which is basically the same, but I guess the teacher wanted it to be more simplified

Thank you so much for taking time to respond tho :D

Guest May 31, 2021