Discussion
Let \(\phi\) and \(\psi\) be two linearly independent solutions of the ordinary differential equation
\[ y'' + (2 - \cos x)\,y = 0, \quad x \in \mathbb{R}. \]
Let \(\alpha, \beta \in \mathbb{R}\) be such that \(\alpha \lt \beta\), \(\phi(\alpha) = \phi(\beta) = 0\) and \(\phi(x) \ne 0\) for all \(x \in (\alpha, \beta)\).
Consider the following statements:
P: \(\phi'(\alpha)\phi'(\beta) \gt 0\).
Q: \(\phi(x)\psi(x) \ne 0\) for all \(x \in (\alpha, \beta)\).
Then
Answer: C
Both P and Q are FALSE.
P: \(\phi\) has consecutive zeros at \(\alpha\) and \(\beta\) and keeps one sign between them; say \(\phi \gt 0\) on \((\alpha, \beta)\). Then \(\phi'(\alpha) \ge 0\) and \(\phi'(\beta) \le 0\), and neither can be zero, since a solution with \(\phi = \phi' = 0\) at a point is identically zero. So \(\phi'(\alpha)\phi'(\beta) \lt 0\), and P is false.
Q: by the Sturm separation theorem, between two consecutive zeros of \(\phi\) the linearly independent solution \(\psi\) has exactly one zero. At that point \(\phi\psi = 0\), so Q is false.
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