A curve is defined on by and

MATHEMATICS · Single correct · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

A curve is defined on −1≤t≤1-1\le t\le 1 by x(t)=t2x(t)=t^{2} and y(t)=t5y(t)=t^{5}. The equation d2ydx2=0\dfrac{d^{2}y}{dx^{2}}=0 is satisfied at t=0t=0. Which statement correctly describes the concavity at this parameter value? (a) t=0t=0 is not an inflection point because dxdt=0\dfrac{dx}{dt}=0, so the slope dydx\dfrac{dy}{dx} is undefined. (b) t=0t=0 is an inflection point because the concavity changes sign, even though dxdt=0.\dfrac{dx}{dt}=0. (c) t=0t=0 is not an inflection point because d2ydx2=152t2\dfrac{d^{2}y}{dx^{2}}=\dfrac{15}{2}t^{2}, which never changes sign. (d) t=0t=0 is an inflection point because d2ydx2=0\dfrac{d^{2}y}{dx^{2}}=0 and dxdt≠0\dfrac{dx}{dt}\ne 0 there.

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