Show that `5x le 8 cos x-2cos 2x le 6x` for `x le x le (pi)/(3)`

Explanation

To show that \( 5x \leq 8 \cos x - 2 \cos 2x \leq 6x \) for \( 0 \leq x \leq \frac{\pi}{3} \), we will analyze the function \( f(x) = 8 \cos x - 2 \cos 2x \) and compare it with the linear functions \( 5x \) and \( 6x \).

### Step 1: Differentiate the function
First, we differentiate the function \( f(x) \):

\[ f'(x) = \frac{d}{dx}(8 \cos x - 2 \cos 2x) \]
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