Consider the equation $y^3 − 3y = x^3 + 1$.

1. What is $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$?

   Step 1(a):

   $\text{Find } \frac{dy}{dx}:$

   $\frac{d}{dx}(y^{3}-3y)=\frac{d}{dx}(x^{3}+1)$

   $3y^{2}\frac{dy}{dx}-3\frac{dy}{dx}=3x^{2}\frac{dx}{dx}$

   $\frac{dy}{dx}(3y^{2}-3)=3x^{2}$

   $\frac{dy}{dx}=\frac{3x^{2}}{3y^{2}-3}=\frac{x^{2}}{y^{2}-1}$

   Step 2(a):

   $\text{Find } \frac{d^{2}y}{dx^{2}}:$

   $\frac{dy}{dx}=\frac{x^{2}}{y^{2}-1}$

   $\frac{d^{2}y}{dx^{2}}=\frac{2x\frac{dx}{dx}(y^{2}-1)-2y\frac{dy}{dx}(x^{2})}{(y^{2}-1)^{2}}=\frac{2x(y^{2}-1)-2y\frac{x^{2}}{y^{2}-1}(x^{2})}{(y^{2}-1)^{2}}$

   You could simplify now, if you choose.

2. Evaluate $\frac { d ^ { 2 } x } { d x ^ { 2 } } | _ { x = 0 }$

   Hint (b):

   $\text{Notice that } \frac{d^{2}y}{dx^{2}},\text{ which you found in part (a),}$

   has both $x$ and $y$ variables.
   So, before you evaluate, use the original equation
   ($y^3 − 3y = x^3 + 1$) to find the $y$-value that corresponds with $x = 0$. It is possible that there will be more than one.

   Step 1(b):

   $y^3 − 3y = (0)^3 + 1$
   $y^3 − 3y − 1 = 0$
   Use a calculator to find the THREE $y$-values that correspond with $x = 0$.

   Step 2(b):

   $\text{Evaluate } \frac{d^{2}y}{dx^{2}}\text{ at those THREE coordinate points.}$

3. Determine all points where the derivative is undefined.

   Hint (c):

   $\text{The derivative, } \frac{dy}{dx} = \frac{x^2}{y^2-1},\text{ will be undefined where the denominator equals 0.}$