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A Fifth-Degree Polynomial and Its Derivatives

This Demonstration displays the graph of the fifth-degree polynomial, , 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 axis, and when the function has a maximum or minimum the first derivative crosses the axis. Check how the second derivative (green) shows the concavity of the function. Why is the fourth derivative a straight line?

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This Demonstration is part of abstractmath.org.
The polynomial and all four derivatives are separated from each other; all of its the roots and those of its derivatives are real and all are shown.
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