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)

Lightning Apr 28, 2019

#1**+1 **

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

CPhill Apr 28, 2019