Mathematics

Problem

A plane curve is given parametrically by

x=t^{2}

and

y=t^{3}-3t

for all real

t

. (a) Express

\dfrac{dy}{dx}

in terms of

t

. (b) Derive an explicit expression for

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

in terms of

t

. Simplify as much as algebraically convenient. (c) Determine all real values of

t

at which the concavity of the curve changes. Show all analytical steps used to reach your conclusion. (d) Give the exact coordinates of every point of inflection on the curve.

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