In Chapter 4, it was discovered that $f ( x ) = \sqrt [ 3 ] { x }$ was not differentiable at $x = 0$.

1. Why does the derivative of $f ( x ) = \sqrt [ 3 ] { x }$ not exist at $x = 0$?

   Hint (a):

   What is the slope of the function at this point? Can you compute it?

2. Is $f ( x ) = \sqrt [ 3 ] { x ^ { 2 } }$ differentiable at $x = 0$? Why or why not?

   Hint (b):

   Graph the function or evaluate the derivative at $0$.

   Answer (b):

   No, the tangent is still vertical (and in this case there is a cusp).

3. What about $f ( x ) = \sqrt [ 3 ] { x ^ { 3 } }$?

   Hint (c):

   Simplify $f(x)$.

4. Explain why there is a point of inflection at $x = 0$ for $f ( x ) = \sqrt [ 3 ] { x }$.

   Hint (d):

   What happens on either side of a point of inflection?What kind of change does a point of inflection signify?

   Answer (d):

   Even though $f^{\prime\prime}(0) =$ DNE, there is still a point of inflection because the graph changes from concave up to concave down at $x = 0$.