For a differentiable parametric curve defined by and

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

Problem

For a differentiable parametric curve defined by

x=f(t)

and

y=g(t)

, a student solves

\dfrac{d^{2}y}{dx^{2}}=0

and obtains

t=t_{0}

. Which of the following best explains the additional condition that must be verified to conclude that the point corresponding to

t_{0}

is an inflection point? (a) Verify that

\dfrac{d^{3}y}{dx^{3}}\neq0

t_{0}

. (b) Verify that

\dfrac{dy}{dx}=0

t_{0}

. (c) Verify that

\dfrac{d^{2}y}{dx^{2}}

changes sign on intervals immediately to either side of

t_{0}

. (d) Verify that

\dfrac{dx}{dt}\neq0

t_{0}