Discussion
Let \(f(x) = (8sinx + 15cosx + 3)^{2} - 15, x \in \mathbb{R}\). Then the maximum value of \(f\) is
Answer: C
The maximum value is 385.
\(8\sin x + 15\cos x\) ranges between \(\pm\sqrt{8^{2} + 15^{2}} = \pm 17\). Adding 3 gives the range [-14, 20] for the expression inside the bracket.
\[ \begin{aligned} \max (8\sin x + 15\cos x + 3)^{2} &= 20^{2} = 400 \\ \max f &= 400 - 15 = 385 \end{aligned} \]
The square is largest at the end of the range farther from zero, which is 20, not -14 (whose square, 196, is smaller). Taking 17 without the 3 gives 274, and \(17^{2} + 3^{2}\) or similar slips give the other options.
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