For a 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 curve defined by

x=f(t)

and

y=g(t)

, the following values are known at

t=2

\frac{dx}{dt}=-4, \qquad \frac{d}{dt}\!\left(\frac{dy}{dx}\right)=6.

Which statement correctly describes the concavity of the curve at the corresponding point? (a) The concavity cannot be determined because

dy/dx

t=2

is unknown. (b) The curve is concave down because

\displaystyle \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}<0

. (c) The curve is concave up because

\dfrac{d}{dt}\bigl(\frac{dy}{dx}\bigr)

is positive. (d) The curve is concave up because both given derivatives are non–zero.