# Determine the x-y coordinates of the points where the following parametric equations will have horizontal or vertical tangents x=t^3-3ty=3t^2-9

Determine the x-y coordinates of the points where the following parametric equations will have horizontal or vertical tangents $$\displaystyle{x}={t}^{{3}}-{3}{t}$$
$$\displaystyle{y}={3}{t}^{{2}}-{9}$$

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Szeteib
$$\displaystyle{x}={t}^{{3}}-{3}{t}$$
$$\displaystyle{y}={3}{t}^{{2}}-{9}$$
$$\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}=\frac{{{6}{t}}}{{{3}{t}^{{2}}-{3}}}$$
for horizontal tangent $$\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}={0}\Rightarrow{t}={0},{P}{\left({0},-{9}\right)}$$
For verticle tangent $$\displaystyle\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}=\infty{o}{t}\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}=\frac{{1}}{{0}}$$
$$\displaystyle\frac{{{6}{t}}}{{{3}{t}^{{2}}-{3}}}=\frac{{1}}{{0}}\Rightarrow{t}=\pm{1}$$
$$\displaystyle{P}{\left(-{2},-{6}\right)}\phi{\left({2},-{6}\right)}$$