What is the smallest distance between the origin and a point on the graph of \(y=\dfrac{1}{\sqrt{2}}\left(x^2-8\right)\)?
+9519 Let \((x, y) = \left(x, \dfrac1{\sqrt2}(x^2 - 8)\right)\) be a point on the graph.
Then the distance between the origin to that point is \(\sqrt{x^2 + \left(\dfrac1{\sqrt 2}(x^2- 8)\right)^2} = \sqrt{x^2 + \dfrac{(x^2 - 8)^2}2} = \sqrt{\dfrac{x^4 - 14x^2 + 64}2}\)
To minimize the distance, we need to minimize \(x^4 - 14x^2 + 64\). Note that \(x^4 - 14x^2 + 64 = (x^2 - 7)^2 + 15 \geq 15\), with equality attained when x = sqrt(7) or x = -sqrt(7). Then the minimum distance is \(\sqrt{\dfrac{15}2} = \dfrac{\sqrt{30}}2\).
