:400px The rate of change of $\theta$ is given by: :$\omega = \dfrac {\d \theta} {\d t}$ From Equation of Astroid: Parametric Form, the point $P = \tuple {x, y}$ is described by the parametric equation: :$\begin {cases} x & = a \cos^3 \theta \\ y & = a \sin^3 \theta \end{cases}$ The rate of change of $y$ is given by: {{begin-eqn}} {{eqn | l = \dfrac {\d y} {\d t}       | r = \dfrac {\d y} {\d \theta} \dfrac {\d \theta} {\d t}       | c = Chain Rule for Derivatives }} {{eqn | r = 3 a \omega \sin^2 \theta \cos \theta       | c = Power Rule for Derivatives, Derivative of Sine Function, Chain Rule for Derivatives }} {{end-eqn}} By Derivative at Maximum or Minimum, when $\dfrac {\d y} {\d t}$ is at a maximum: :$\dfrac {\d^2 y} {\d t^2} = 0$ Thus: {{begin-eqn}} {{eqn | l = \dfrac {\d^2 y} {\d t^2}       | r = \map {\dfrac \d {\d \theta} } {3 a \omega \sin^2 \theta \cos \theta} \dfrac {\d \theta} {\d t}       | c = Chain Rule for Derivatives }} {{eqn | r = 3 a \omega^2 \paren {2 \sin \theta \cos^2 \theta - \sin^3 \theta}       | c = Product Rule for Derivatives and others }} {{end-eqn}} Hence: {{begin-eqn}} {{eqn | l = \dfrac {\d^2 y} {\d t^2}       | r = 0       | c =  }} {{eqn | ll= \leadsto       | l = 3 a \omega^2 \paren {2 \sin \theta \cos^2 \theta - \sin^3 \theta}       | r = 0       | c =  }} {{eqn | ll= \leadsto       | l = 2 \sin \theta \cos^2 \theta       | r = \sin^3 \theta       | c =  }} {{end-eqn}} We can assume $\sin \theta \ne 0$ because in that case $\theta = 0$ and so $\dfrac {\d y} {\d t} = 0$. Thus when $\sin \theta = 0$, $y$ is not a maximum. So we can divide by $\sin \theta$ to give: {{begin-eqn}} {{eqn | l = 2 \cos^2 \theta       | r = \sin^2 \theta       | c =  }} {{eqn | ll= \leadsto       | l = \tan^2 \theta       | r = 2       | c = Tangent is Sine divided by Cosine }} {{eqn | ll= \leadsto       | l = \tan \theta       | r = \sqrt 2       | c =  }} {{end-eqn}} We have: {{begin-eqn}} {{eqn | l = x       | r = a \cos^3 \theta       | c =  }} {{eqn | l = y       | r = a \sin^3 \theta       | c =  }} {{eqn | ll= \leadsto       | l = \frac y x       | r = \tan^3 \theta       | c = Tangent is Sine divided by Cosine }} {{eqn | r = 2^{3/2}       | c =  }} {{eqn | ll= \leadsto       | l = y       | r = 2^{3/2} x       | c =  }} {{end-eqn}} From Equation of Astroid: Cartesian Form: :$x^{2/3} + y^{2/3} = a^{2/3}$ Hence: {{begin-eqn}} {{eqn | l = x^{2/3} + \paren {2^{3/2} x}^{2/3}       | r = a^{2/3}       | c =  }} {{eqn | ll= \leadsto       | l = x^{2/3} \paren {1 + 2}       | r = a^{2/3}       | c = simplifying }} {{eqn | ll= \leadsto       | l = x       | r = \paren {\frac {a^{2/3} } 3}^{3/2}       | c =  }} {{eqn | ll= \leadsto       | l = x       | r = a \paren {\frac 1 3}^{3/2}       | c =  }} {{end-eqn}} Similarly: {{begin-eqn}} {{eqn | l = \frac y x       | r = 2^{3/2}       | c =  }} {{eqn | ll= \leadsto       | l = x       | r = \frac 1 {2^{3/2} } y       | c =  }} {{eqn | l = \paren {\frac 1 {2^{3/2} } y}^{2/3} + y^{2/3}       | r = a^{2/3}       | c =  }} {{eqn | ll= \leadsto       | l = y^{2/3} \paren {1 + \frac 1 2}       | r = a^{2/3}       | c = simplifying }} {{eqn | ll= \leadsto       | l = y^{2/3}       | r = \frac 2 3 a^{2/3}       | c = simplifying }} {{eqn | ll= \leadsto       | l = y       | r = \paren {2 \frac {a^{2/3} } 3}^{3/2}       | c =  }} {{eqn | ll= \leadsto       | l = x       | r = a \paren {\frac 2 3}^{3/2}       | c =  }} {{end-eqn}} {{qed}}