Given: $y^{2} = x − x^{3}$ :

1. Write an equation for $\frac { d y } { d x }$.

   Hint (a):

   Implicit differentiation.

2. For what value of $y$ is there a vertical tangent to the graph?

   Hint (b):

   Determine all $y$-values in which the denominator of the derivative is $0$.

3. For what values of $x$ are there vertical tangents to the graph?

   Hint (c):

   Use the original function to determine the corresponding $x$-value for each $y$-value you found in part (b).

   Hint (c):

   There will be three values of $x$ that work. That means there will be three coordinate points in which the slope is vertical: ( ____, $0$ ), ( ____, $0$ ) and ( ____, $0$ ). (Obviously, this is NOT a function!)

4. Write an equation for $\frac { d ^ { 2 } y } { d x ^ { 2 } }$.

   Hint (d):

   The second derivative must be written in terms of $x$ and $y$ only.

   $\text{If }\frac{dy}{dx}\text{ appears in the second derivative, substitute it with the value you found in part (a).}$