Mathematics

Problem

Let

a

be a non-zero constant. A curve is given by

x = t^{3} - 3at

y = t^{4} - 4at^{2} + 2a^{2}

, where

t \in \mathbb{R}

. a. (i) Obtain expressions for

\dfrac{dy}{dx}

and

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

in terms of

t

and

a

. b. (ii) Show that the curve possesses points of inflection only when

t^{2} = a

, and determine the corresponding coordinates in terms of

a

. c. (iii) For

a>0

, find the equation of the tangent to the curve at each point of inflection and state whether the curve changes from concave upward to concave downward or vice-versa at those points.

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