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What is the smallest distance between the origin and a point on the graph of y = 1/2*(x^2 - 3)?

 Dec 14, 2021
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Express the distance as a function. The distance between any point (x,y) and the origin is √(x^2 + y^2). Since y = 1/2(x^2-3), we have to minimize √(x^2 + (1/2(x^2-3))^2), or 1/2 √(x^4 - 2 x^2 + 9). To minimize the expression inside the radical, we let a=x^2, and find the minimum value/vertex of the parabola that results from its graph. Completing the square, we have a^2-2a+9 --> a^2-2a+1 + 8, or (a-1)^2 + 8. This makes the vertex (1,-8), so 1 = x^2, and x = ±1. Inputting this into our expression (x^2-1)^2 + 8, we get 8, so our answer is 1/2*√8 = 1/2 * 2√2 = √2.

 Dec 14, 2021

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