(Solution) - a Investigate the family of polynomials given by the equation -(2025 Original AI-Free Solution)

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Academic Level: Undergrad. (yrs 3-4)

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(a) Investigate the family of polynomials given by the equation f(x) = 2x3 + cx2 + 2x. For what values of does the curve have maximum and minimum points?
(b) Show that the minimum and maximum points of every curve in the family lie on the curve y = x ? x3. Illustrate by graphing this curve and several members of the family.