What is the smallest distance between the origin and a point on the graph of y = 1/2*(x^2 - 8)?
+2668 We want to minimize \(\sqrt{x^2+ y^2} \)
Substituting \(y = {1 \over 2}\times(x^2 - 8)\), we get \( \sqrt{x^2 + {x^4 \over 4} - 4x^2 + 16}\), which simplifies to \( \sqrt{{x^4 \over 4} - 3x^2 + 16}\).
Now, let \(z = x^2\). We have \( \sqrt{{z^2 \over 4} - 3z + 16}\). Because this is a quadratic, the minimum occurs at \(-{b \over 2a} = -{3 \over 2 \times {0.25}} = 6\).
Substituting this in gives us \( \sqrt{{6^2 \over 4} - 3\times 6^2 + 16} =\color{brown}\boxed{ \sqrt{7}}\)
