<p>The number of critical points of \[f(x) = \max\{\sin x, \cos x\}, \quad x \in (-2\pi, 2\pi)\] is</p>
Step-by-Step Solution
Key Concept: Critical points of piecewise maximum functions occur at intersection points where the maximum switches from one function to another, and at local extrema of the individual functions.
<p><strong>Step 1:</strong> The function $f(x) = \max\{\sin x, \cos x\}$ has critical points where:</p><p>• $\sin x = \cos x$ (intersection points where the maximum changes)</p><p>• Where $\sin x$ or $\cos x$ individually have extrema and form the boundary</p><p><strong>Step 2:</strong> $\sin x = \cos x$ gives $\tan x = 1$, so $x = \frac{\pi}{4} + n\pi$</p><p><strong>Step 3:</strong> In the interval $(-2\pi, 2\pi)$, the solutions are:</p><p>$$x = -\frac{7\pi}{4}, -\frac{3\pi}{4}, \frac{\pi}{4}, \frac{5\pi}{4}$$</p><p><strong>Step 4:</strong> Additionally, we have critical points where either $\sin x$ or $\cos x$ reach local maxima within the max function:</p><p>$$x = -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}$$</p><p><strong>Step 5:</strong> Counting all critical points: 7 critical points in total.</p><p>∴ Answer is (C) 7</p>
Correct Answer: C