Discussion
The value of \(C\) in Rolle's theorem for the function \(f(x) = \cos 4x\) in \([0, \pi]\) is:
Answer: D
Rolle's theorem needs \(f(0) = f(\pi)\) and a point \(c\) in \((0, \pi)\) where \(f'(c) = 0\). Here \(f(0) = \cos 0 = 1\) and \(f(\pi) = \cos 4\pi = 1\), so the theorem applies.
\[ f'(x) = -4\sin 4x = 0 \]
\[ 4x = k\pi \Rightarrow x = \frac{k\pi}{4} \]
Inside the open interval this gives \(\pi/4\), \(\pi/2\) and \(3\pi/4\). Of the values offered, only \(\pi/4\) is one of them. The values \(0\) and \(\pi\) are the end points, not interior points, and \(\pi/3\) gives \(f'(\pi/3) = -4\sin(4\pi/3) \neq 0\). The answer is \(\pi/4\), option D.
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