The smallest distance between the origin and a point on the graph of y = 1/3*x^2 - 9 can be expressed as a. Find a^2.
+128475 Let the point we are looking for be ( m , (1/3)m^2 -9)
We can minimize the square of the distance formula
D^2 = (m)^2 + [ (1/3) m^2 - 9 ]^2
D^2 = m^2 + (1/9)m^4 - 6m^2 + 81
D^2 = (1/9)m^4 - 5m^2 + 81
Take the derivative of the functtion and set to 0
D' ^2 = (4/9)m^3 - 10m = 0
(4/9)m^3 - 10m = 0
m [ ( 4//9)m^2 - 10 ] = 0
The second factor dives us what we need
(4/9)m^2 - 10 = 0
(4/9)m^2 = 10
m^2 = 45/2
We have two possible values m = sqrt (45/2) or m = -sqrt (45/2)
Either will give us D^2 = a^2 = (1/9)(sqrt (45/2))^4 -5 (sqrt (45/2 ))^2 + 81 = 99/4
See the graph here : https://www.desmos.com/calculator/pleksd640o
