Find the smallest distance between the origin and a point on the parabola y = x^2 -2.
+128408 y = x^2 - 2
Let the point on the parabola be ( x , x^2 - 2)
Using the distance formula we have that
D = [ x^2 + ( x^2 - 2)^2 ]^(1/2) simplify
D = [ x^2 + x^4 - 4x^2 + 4] ^(1/2)
D = [ x^4 - 3x^2 + 4 ) ^(1/2) take the derivative of this and set to 0
D' = [ 4x^3 + 6x ] / [ x^4 - 3x^2 + 4 ] ^(1/2) = 0
So
4x^3 + 6x = 0
2x (2x^2 - 3) = 0
2x = 0 2x^2 - 3 = 0
x = 0 reject 2x^2 = 3
x^2 = 3/2
x = ±sqrt (3/2)
When x = sqrt (3/2) then y = [sqrt (3/2)] ^2 - 2 = -1/2
When x = -sqrt (3/2) y has the same value = -1/2
So....the shortest distance in either case is
sqrt ( 3/2 + 1/2) = sqrt (2)
See the graph, here : https://www.desmos.com/calculator/i36wydpg70
