Question

What is the smallest distance between the origin and a point on the graph of $y=(x^2-3)/(sqrt{2})

Answer 1

I read that as
y = (x^2 - 3)/√2

dy/dx = (1/√2)(2x) = 2x/√2

let that closest point be P(x,y)
so the slope of the tangent at P = 2x/√2

slope of PO = y/x
at the closest point, PO is perpendicular to the tangent at P
then,
y/x = -√2/(2x)
y = - √2/2

for the x:
(x^2-3)/√2 = -√2/2
x^2 - 3 = -1
x^2 = 2
x = ± √2

looking at my sketch , I will pick
(√2 , -√2/2)