math

Find the smallest distance between the origin and a point on the parabola y = x^2 - 2.

(where I have distance from origin, I should have put distance squared from origin!)

Origin is: (0, 0)

Point in parabola: (x, y)

Distance: sqrt(x^2 + y^2)

sqrt(x^2 + y^2)

sqrt(x^2 + (x^2 - 2)^2)

sqrt(x^2 + x^4 - 4x^2 + 4)

sqrt(x^4 - 3x^2 + 4)

Now, we have to find the smallest positive value of x^4 - 3x^2 + 4. 

Not quite sure how to do that. 

=^._.^=

Can use calculas as in #1 above, or altermatively completing the square works.

\(\displaystyle x^{4}-3x^{2}+4 = x^{4}-3x^{2}+(9/4) + 4 -(9/4)\\ = (x^{2}-3/2)^{2}+7/4,\)

so the minimum distance is \(\displaystyle \sqrt{7/4}\) from either \(\displaystyle x=\pm\sqrt{3/2}.\)