Mathematics

Problem

Consider the parametric equations

x=t^{3}-3t^{2}, \qquad y=t^{2}-2t,\qquad -2\le t\le3.

Let

I

be the set of

x

-values satisfying

-1\le x\le1

. (a) Determine the corresponding set of

t

-values that map into

I

. Present your answer as one or more intervals. (b) On each

t

-interval found in part (a), evaluate the sign of

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

. Show analytical work supporting each sign determination. (c) State the

x

-intervals over which the curve is concave up and the

x

-intervals over which it is concave down, restricted to

I

. Justify your conclusions.