find the maximum and minimum values of 4/(2sinx+9)

To find the maximum and minimum values of the function \( f(x) = \frac{4}{2\sin(x) + 9} \), we follow these steps:

### Step 1: Analyze the Denominator
The function depends on the term \( 2\sin(x) + 9 \) in the denominator. Since \( \sin(x) \) varies between \(-1\) and \( 1 \) for all real \( x \):

\[
-1 \leq \sin(x) \leq 1
\]

Multiply this inequality by 2:

\[
-2 \leq 2\sin(x) \leq 2
\]

Add 9 to all sides:

\[
7 \leq 2\sin(x) + 9 \leq 11
\]

So, the denominator \( 2\sin(x) + 9 \) varies between 7 and 11.

### Step 2: Determine the Behavior of the Function
The function \( f(x) = \frac{4}{2\sin(x) + 9} \) is the reciprocal of the denominator. Since the denominator is smallest when \( 2\sin(x) + 9 \) is 7 and largest when \( 2\sin(x) + 9 \) is 11:

- \( f(x) \) will be **maximum** when the denominator is minimum (7).
- \( f(x) \) will be **minimum** when the denominator is maximum (11).

### Step 3: Calculate Maximum and Minimum Values
1. **Maximum Value:**

\[
f_{\text{max}} = \frac{4}{7} \approx 0.571
\]

2. **Minimum Value:**

\[
f_{\text{min}} = \frac{4}{11} \approx 0.364
\]

### Conclusion:
- The **maximum value** of the function is \( \frac{4}{7} \approx 0.571 \).
- The **minimum value** of the function is \( \frac{4}{11} \approx 0.364 \).

Would you like more details on this process, or do you have any questions?

### Relative Questions:
1. How do you find the critical points of a trigonometric function?
2. Why does the function achieve its extrema at the boundary values of \( \sin(x) \)?
3. Can you explain the effect of modifying the amplitude and phase of the sine function on \( f(x) \)?
4. How would the maximum and minimum values change if the constant 9 were different?
5. What happens to the function \( f(x) = \frac{4}{2\sin(x) + 9} \) as \( x \) approaches infinity?

### Tip:
Understanding how the sine function varies and affects the overall expression is key to solving many trigonometric problems.

Discover how to find the maximum and minimum values of the function f(x) = 4/(2sin(x) + 9) with a step-by-step solution. Learn about trigonometric functions, reciprocal functions, and function analysis. Suitable for advanced high school students and college-level learners.

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