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.