\(\text{Let $x_1,$ $x_2,$ $\dots,$ $x_9$ be real numbers such that \[\cos x_1 + \cos x_2 + \dots + \cos x_9 = 0.\]Find the maximum value of $\cos 3x_1 + \cos 3x_2 + \dots + \cos 3x_9.$}\)
+9519 \(\displaystyle \sum_{k = 1}^9 \cos 3x_k\\ = \displaystyle \sum_{k = 1}^9 \left(4\cos^3 x_k-3\cos x_k\right)\\ = 4\displaystyle \sum_{k = 1}^9 \cos^3 x_k - 3\sum_{k = 1}^9 \cos x_k\\ =4\displaystyle \sum_{k = 1}^9 \cos^3 x_k\\ \le 4 \left(\displaystyle \sum_{k = 1}^9 \cos x_k\right)^3\\ = 0\\ \therefore \text{Maximum value is 0.}\)
.
