Consider the parametric curve defined by and for

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

Consider the parametric curve defined by x=t3−3tx = t^{3}-3t and y=t2−2ty = t^{2}-2t for t∈[−2,3]t\in[-2,3]. Answer the following: (i) Find d2ydx2\dfrac{d^{2}y}{dx^{2}} in terms of tt and simplify your result. (ii) Determine all real values of tt in the interval [−2,3][-2,3] at which d2ydx2=0\dfrac{d^{2}y}{dx^{2}}=0. For each such value, test the sign of d2ydx2\dfrac{d^{2}y}{dx^{2}} immediately to its left and right. (iii) Hence list the sub-intervals of tt on which the curve is concave upward and those on which it is concave downward. State the coordinates of every point of inflection that lies on the curve in the given interval.

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