Given: y² = x − x³ :

1. Find $\frac { d y } { d x }$.

   Hint (a):

   Implicit differentiation.

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

   Hint (b):

   Find all y-values in which the denominator of the derivative = 0.

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

   Hint 1(c):

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

   Hint 2(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. Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$.

   Hint (d):

   The 2nd-derivative must be written in terms of x and y only.