If `f(x)= |sin x + 2/(3+sin x)+b|` and f(b) denote the maximum of the function ` forall x in R`, then the minimum value of f(b) is

Options

A. `3/2`
B. `0`
C. `1/2`
D. `-3/4`

Answer

To solve the problem, we need to find the minimum value of the function \( f(b) = |g(b)| \), where \( g(x) = \sin x + \frac{2}{3 + \sin x} \). We will follow these steps:

### Step 1: Analyze the function \( g(x) \)

The function \( g(x) \) is defined as:
\[ g(x) = \sin x + \frac{2}{3 + \sin x} \]
To find the maximum and minimum values of \( g(x) \), we need to analyze the behavior of this function.

### Step 2: Determine the range of \( \sin x \)

The sine function oscillates between -1 and 1:
\[ -1 \leq \sin x \leq 1 \]

### Step 3: Substitute the extreme values of \( \sin x \)

1. **When \( \sin x = -1 \)**:
\[ g(-1) = -1 + \frac{2}{3 - 1} = -1 + 1 = 0 \]

2. **When \( \sin x = 1 \)**:
\[ g(1) = 1 + \frac{2}{3 + 1} = 1 + \frac{2}{4} = 1 + \frac{1}{2} = \frac{3}{2} \]

### Step 4: Determine the maximum and minimum values of \( g(x) \)

From the calculations above, we see that:
- The minimum value of \( g(x) \) is \( 0 \) (when \( \sin x = -1 \)).
- The maximum value of \( g(x) \) is \( \frac{3}{2} \) (when \( \sin x = 1 \)).

### Step 5: Express \( f(b) \)

The function \( f(b) \) can be expressed as:
\[ f(b) = |g(b)| = |g(x) + b| \]
To find the maximum value of \( f(b) \), we need to consider the maximum value of \( g(x) \):
\[ f(b) = |g(x)| + b \]
The maximum value of \( g(x) \) is \( \frac{3}{2} \), so:
\[ f(b) = \frac{3}{2} + b \]

### Step 6: Find the minimum value of \( f(b) \)

To find the minimum value of \( f(b) \), we need to minimize \( |g(b)| + b \). The minimum value of \( |g(b)| \) is \( 0 \), which occurs when \( b = -\frac{3}{2} \):
\[ f(b) = 0 + b = b \]
Thus, the minimum value of \( f(b) \) occurs when:
\[ b = -\frac{3}{2} \]

### Conclusion

The minimum value of \( f(b) \) is:
\[ \text{Minimum value of } f(b) = 0 \]