Mathematics

Problem

Consider the parametric curve

x=t^{3}-3t

and

y=t^{4}-4t^{2}+5

. a. (i) Show that

\dfrac{dy}{dx}=\dfrac{4t^{3}-8t}{3t^{2}-3}

. b. (ii) Determine all real values of

t

for which the tangent to the curve is horizontal. c. (iii) Determine all real values of

t

for which the tangent to the curve is vertical. d. (iv) For each value of

t

obtained in parts (ii) and (iii), find the corresponding coordinates of the points on the curve.