Mathematics

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

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

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

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