Mathematics

Problem

Consider the curve defined parametrically by

x = t^2 + 1

and

y = t^3 - 3t

, where

t \in \mathbb{R}

. Answer the following questions: a. (i) Find

\dfrac{dy}{dx}

and

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

in terms of

t

. b. (ii) Determine the nature of concavity of the curve at the points corresponding to

t = -1

and

t = 1

. c. (iii) Using your result in part (ii), comment on whether the point corresponding to

t = 0

is an inflection point. Justify your answer.