Mathematics

Problem

A curve is given by

x(t)=t^{2}+1, \qquad y(t)=t^{3}-t,

for all real numbers

t

. The curve passes through the point

Q=(2,0)

. (a) Show that

t=1

corresponds to point

Q

. (b) Derive an explicit formula for

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

in terms of

t

for this curve. (c) Evaluate the second derivative at

t=1

. (d) State and justify whether the curve is concave up or concave down at

Q

. Include a brief written explanation that references your work in parts (b) and (c).