For the function, find all points of inflection or determine that no such points exist.

*f* (*x*) = *x*^{1/3}

Answer

*f* (*x*) = *x*^{1/3}

We find the first two derivatives:

This function is undefined when *x* = 0. Since *x* = 0 is in the domain of *f*, this is a possible inflection point. To determine if *x* = 0 is a real inflection point, we need to inspect the sign of *f *" to either side.

When *x* is greater than zero,

is also greater than zero, so

is negative.

When *x* is less than zero,

is also less than zero because of the odd numbers in the fractional exponent (the cube root of a negative number is negative, and raising a negative number to the 5th power is negative). Therefore

is positive.

Since *f *" changes sign at *x* = 0, this is an inflection point. Here's the graph: