Mathematics

Problem

For the parametric equations

x = t^{2}+1

and

y = t^{3}-3t

(

t\in\mathbb{R}

), respond to the tasks below: a. (i) Find all values of

t

at which the tangent to the curve is horizontal and all values where it is vertical. b. (ii) Determine the value(s) of

t

for which

\dfrac{d^{2}y}{dx^{2}}=0

. c. (iii) By analysing the sign of

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

on either side of each candidate obtained in part (ii), identify the actual point(s) of inflection, giving their coordinates. d. (iv) Comment on whether any point of inflection coincides with a horizontal or vertical tangent for this curve, giving reasons for your answer.

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