Theorem: Maximum Rate of Change of Y Coordinate of Astroid

Let $C_1$ and $C_2$ be the epicycle and deferent respectively of an astroid $H$.
Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin.
Let the center $C$ of $C_1$ move at a constant angular velocity $\omega$ around the center of $C_2$.
Let $P$ be the point on the circumference of $C_1$ whose locus is $H$.
Let $C_1$ be positioned at time $t = 0$ so that $P$ its point of tangency to $C_2$, located on the $x$-axis.
Let $\theta$ be the angle made by $OC$ to the $x$-axis at time $t$.
Then the maximum rate of change of the $y$ coordinate of $P$ in the first quadrant occurs when $P$ is at the point where:
:$x = a \paren {\dfrac 1 3}^{3/2}$
:$y = a \paren {\dfrac 2 3}^{3/2}$

Proof:
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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 the Interior Extremum Theorem, 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}}