+1245 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)
+129852 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
