Consider a smooth curve defined parametrically by and on an open interval

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

Problem

Consider a smooth curve defined parametrically by

x=f(t)

and

y=g(t)

on an open interval

I

. Assume

\dfrac{dx}{dt}\neq 0

throughout

I

. (a) Describe how the sign of

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

affects the concavity of the curve in the

xy

–plane. (b) Explain why the condition

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

(or being undefined) at

t=t_{0}\in I

is necessary but not sufficient for the point

(f(t_{0}),g(t_{0}))

to be an inflection point. (c) A particular parametric graph has

\dfrac{d^{2}y}{dx^{2}}\!>\!0

for

t<2

and

\dfrac{d^{2}y}{dx^{2}}\!<\!0

for

t>2

. Without computing any derivatives, determine whether

(f(2),g(2))

is an inflection point and justify your answer in terms of concavity.