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Numerade Educator

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Problem 38 Medium Difficulty

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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