Question

Let $f(x) = \frac{10}{7 + 4\sin x + 3\cos x}, x \in \mathbb{R}$. Then the range of the function $f$ is

• A. $[\frac{5}{7}, 5]$
• B. $[\frac{5}{7}, \frac{10}{7}]$
• C. $[\frac{5}{6}, 5]$
• D. $[\frac{5}{3}, 5]$
• E. $[\frac{5}{3}, \frac{10}{3}]$

Hint:

For \( a \sin x + b \cos x \), remember the Pythagorean triplets like (3, 4, 5). This allows you to immediately identify the range as $\pm 5$ without lengthy calculations.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
To find the range of a function where the variable is inside a trigonometric expression in the denominator, we must first determine the maximum and minimum values of that denominator.
Key Formula or Approach:
The range of the expression \( a \sin x + b \cos x \) is \( [-\sqrt{a^2 + b^2}, \sqrt{a^2 + b^2}] \).

Step 2: Detailed Explanation:
Consider the expression in the denominator: \( D = 7 + 4\sin x + 3\cos x \).
The range of the part \( 4\sin x + 3\cos x \) is:
\[ [-\sqrt{4^2 + 3^2}, \sqrt{4^2 + 3^2}] = [-\sqrt{16 + 9}, \sqrt{16 + 9}] = [-5, 5] \]
Adding 7 to find the range of the whole denominator:
\[ \text{Minimum denominator} = 7 - 5 = 2 \]
\[ \text{Maximum denominator} = 7 + 5 = 12 \]
Now, calculate the range of the function \( f(x) = \frac{10}{D} \):
\[ \text{Maximum value of } f(x) = \frac{10}{\text{Minimum denominator}} = \frac{10}{2} = 5 \]
\[ \text{Minimum value of } f(x) = \frac{10}{\text{Maximum denominator}} = \frac{10}{12} = \frac{5}{6} \]
Thus, the range of the function is \( [\frac{5}{6}, 5] \).

Step 3: Final Answer:
The range is \( [\frac{5}{6}, 5] \).