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Investigate the family of curves given by the equation $ f(x) = x^4 + cx^2 + x $. Start by determining the transitional value of $ c $ at which the number of inflection points changes. Then graph several members of the family to see what shapes are possible. There is another transitional value of $ c $ at which the number of critical numbers changes. Try to discover it graphically. Then prove what you have discovered.
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Calculus 1 / AB
Calculus 2 / BC
Chapter 4
Applications of Differentiation
Section 6
Graphing with Calculus and Calculators
Derivatives
Differentiation
Volume
Missouri State University
Harvey Mudd College
University of Nottingham
Boston College
Lectures
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Investigate the family of …
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Describe how the graph of …
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