Originally Posted by

**HallsofIvy** The two parts of the function, y= 4 and y= 16x+ 4, are both straight lines so the **second** derivatives are 0 for x< 0 and x> 0. But that does NOT mean the second derivative exists **at** x= 0! The **first** derivative is 0 for x< 0 and 16 for x> 0 but does not exist **at** x= 0 so the second derivative cannot exist at x= 0.

The first derivative of this function is "4 for x< 0, 16 for x> 0, does not exist at x= 0". The second derivative (and all higher derivatives) is "0 for x< 0 or x> 0, does not exist at x= 0".