+646 What is the smallest distance between the origin and a point on the graph of y = 1/sqrt2 (x^2 - 3)?
+128406 Let the point be [ x, (1/√2) ( x^2 - 3) ]
D = √[ x^2 + (1/2) (x^4 - 6x^2 + 9) ]
D ' = (1/2) [x^2 + (1/2) (x^2 - 6x^2 + 9)]^(-1/2) [2x + (1/2)(4x^3 - 12x ) ]
Set this to 0 and solve for x and we get that
2x + (1/2) (4x^3 - 12x) = 0
2x + 2x^3 - 6x = 0
2x^3 - 4x = 0
2x ( x*2 - 2) = 0
x = √2
So
D = √ [2 + (1/2) (4 - 12 + 9 ) ] = √[2 + 1/2 ] = √ [2.5] units ≈ 1.58 units
