A Fifth-Degree Polynomial and Its Derivatives

This Demonstration displays the graph of the fifth-degree polynomial,

f(x)=0.01

5

x

-0.015

4

x

-0.21

3

x

+0.2

2

x

+0.5x-0.5

, and up to four of its derivatives. As you move the slider across, note how when the function (blue curve) goes down, the first derivative (orange) is below the

x

axis, and when the function has a maximum or minimum the first derivative crosses the

x

axis. Check how the second derivative (green) shows the concavity of the function. Why is the fourth derivative a straight line?