Suppose is strictly increasing on an interval and is twice differentiable

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

Problem

Suppose

x(t)

is strictly increasing on an interval and

y(t)

is twice differentiable. At a value

t=c

, the curve is concave upward. Which of the following inequalities in terms of derivatives with respect to

t

is sufficient to conclude this concavity at

t=c

? (a)

\displaystyle \frac{dy}{dx}\bigg|_{t=c}>0

(b)

\displaystyle \frac{d}{dt}\!\left(\frac{dy}{dx}\right)\bigg|_{t=c}<0

(c)

\displaystyle \frac{d}{dt}\!\left(\frac{dy}{dx}\right)\bigg|_{t=c}>0

(d)

\displaystyle \frac{d^{2}y}{dt^{2}}\bigg|_{t=c}>0