A parabol is very useful for analyzing minimization subject; apply it to anything from micro to milimicro; it can help you to find the minimum point after which the curve moves upwards.
On a curve it is not possible to find one point only so you need to have the tangent line through derivatives, which will give you the minimum. Think about y= xsquare + 2x + 1; the derivative will be 2x + 2 = 0 and the curve will be minimized at the point where it meets the tangent, which will be for x = -1.
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