  ### Home > APCALC > Chapter 4 > Lesson 4.2.4 > Problem4-90

4-90.

Earlier in this chapter, 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 $x = 0$?

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

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

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

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

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

Simplify $f\left(x\right)$.

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

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

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