Activity 3.3.4.
Consider the family of functions given by \(h(x) = x^2 + \cos(kx)\text{,}\) where \(k\) is an arbitrary positive real number.
(a)
Use a graphing utility to sketch the graph of \(h\) for several different \(k\)-values, including \(k = 1,3,5,10\text{.}\) Plot \(h(x) = x^2 + \cos(3x)\) on the axes provided. What is the smallest value of \(k\) at which you think you can see (just by looking at the graph) at least one inflection point on the graph of \(h\text{?}\)
(b)
Explain why the graph of \(h\) has no inflection points if \(k \le \sqrt{2}\text{,}\) but infinitely many inflection points if \(k \gt \sqrt{2}\text{.}\)
(c)
Explain why, no matter the value of \(k\text{,}\) \(h\) can only have finitely many critical numbers.
