Discussion
Let \(f(x) = \dfrac{10}{7 + 4\sin x + 3\cos x}\), \(x \in \mathbb{R}\). Then the range of the function \(f\) is
Answer: C
The range is \(\left[\frac{5}{6}, 5\right]\).
The expression \(4\sin x + 3\cos x\) can be written as \(5\sin(x + \phi)\), so it takes every value from \(-5\) to \(5\), since \(\sqrt{4^{2} + 3^{2}} = 5\).
\[ \begin{aligned} 7 - 5 &\le 7 + 4\sin x + 3\cos x \le 7 + 5 \\ 2 &\le \text{denominator} \le 12 \\ \tfrac{10}{12} &\le f(x) \le \tfrac{10}{2} \end{aligned} \]
So \(f\) runs from \(\frac{5}{6}\), when the denominator is largest, to 5, when it is smallest. Every other option has a different lower bound, which would need a different maximum value of the denominator than 12.
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