Activity 3.4.2.
Consider the family of functions defined by \(p(x) = x^3 - ax\text{,}\) where \(a \ne 0\) is an arbitrary constant.
(a)
Find \(p'(x)\) and determine the critical numbers of \(p\text{.}\) How many critical numbers does \(p\) have?
(b)
Construct a first derivative sign chart for \(p\text{.}\) What can you say about the overall behavior of \(p\) if the constant \(a\) is positive? Why? What if the constant \(a\) is negative? In each case, describe the relative extremes of \(p\text{.}\)
(c)
Find \(p''(x)\) and construct a second derivative sign chart for \(p\text{.}\) What does this tell you about the concavity of \(p\text{?}\) What role does \(a\) play in determining the concavity of \(p\text{?}\)
(d)
Without using a graphing utility, sketch and label typical graphs of \(p(x)\) for the cases where \(a\gt 0\) and \(a \lt 0\text{.}\) Label all inflection points and local extrema.
