for ` theta in [ 0, pi ] ,` let ` f( theta) = sin ( cos theta) and g ( theta) = cos ( sin theta ) . ` Let a = ` underset( 0 le theta le pi ) (" max") f ( theta) , b= underset(0 le theta le pi )( min) f( theta), c = underset(0 le theta le pi ) ( " max") g( theta) and d= underset( 0 le theta le pi )( min g( theta)` . the correct inequalities satisfied by ` a,b,c,d are

Options

A. `b lt d lt c lt a `
B. ` d lt b lt a lt c `
C. `b lt d lt a lt c`
D. `b lt a lt d lt c`

Answer

To solve the problem, we need to analyze the functions \( f(\theta) = \sin(\cos(\theta)) \) and \( g(\theta) = \cos(\sin(\theta)) \) over the interval \( \theta \in [0, \pi] \) to find the maximum and minimum values of each function.

### Step 1: Analyze \( f(\theta) \)

1. **Find the derivative of \( f(\theta) \)**:
\[ f'(\theta) = \frac{d}{d\theta} \sin(\cos(\theta)) = \cos(\cos(\theta)) \cdot (-\sin(\theta)) = -\sin(\theta) \cos(\cos(\theta)) \]

2. **Determine the behavior of \( f'(\theta) \)**:
- The term \(-\sin(\theta)\) is non-positive for \( \theta \in [0, \pi] \) (it is zero at \( \theta = 0 \) and \( \theta = \pi \)).
- Thus, \( f'(\theta) \leq 0 \) implies that \( f(\theta) \) is a decreasing function on the interval.

3. **Find maximum and minimum values**:
- Maximum occurs at \( \theta = 0 \):
\[ f(0) = \sin(\cos(0)) = \sin(1) \quad \text{(let this be } a = \sin(1)\text{)} \]
- Minimum occurs at \( \theta = \pi \):
\[ f(\pi) = \sin(\cos(\pi)) = \sin(-1) = -\sin(1) \quad \text{(let this be } b = -\sin(1)\text{)} \]

### Step 2: Analyze \( g(\theta) \)

1. **Find the derivative of \( g(\theta) \)**:
\[ g'(\theta) = \frac{d}{d\theta} \cos(\sin(\theta)) = -\sin(\sin(\theta)) \cdot \cos(\theta) \]

2. **Determine the behavior of \( g'(\theta) \)**:
- The term \(-\sin(\sin(\theta))\) is non-negative for \( \theta \in [0, \pi] \) (it is zero at \( \theta = 0 \) and \( \theta = \pi \)).
- Thus, \( g'(\theta) \geq 0 \) implies that \( g(\theta) \) is an increasing function on the interval.

3. **Find maximum and minimum values**:
- Maximum occurs at \( \theta = 0 \):
\[ g(0) = \cos(\sin(0)) = \cos(0) = 1 \quad \text{(let this be } c = 1\text{)} \]
- Minimum occurs at \( \theta = \frac{\pi}{2} \):
\[ g\left(\frac{\pi}{2}\right) = \cos(\sin\left(\frac{\pi}{2}\right)) = \cos(1) \quad \text{(let this be } d = \cos(1)\text{)} \]

### Step 3: Compare the values \( a, b, c, d \)

- We have:
- \( a = \sin(1) \)
- \( b = -\sin(1) \)
- \( c = 1 \)
- \( d = \cos(1) \)

### Step 4: Establish inequalities

1. **Comparing \( a \) and \( b \)**:
- Since \( a = \sin(1) > 0 \) and \( b = -\sin(1) < 0 \), we have \( b < a \).

2. **Comparing \( c \) and \( d \)**:
- Since \( c = 1 \) and \( d = \cos(1) \) (where \( \cos(1) < 1 \)), we have \( d < c \).

3. **Final comparison**:
- Since \( b < 0 < d < c \) and \( b < a \), we can summarize:
\[ b < d < a < c \]

### Conclusion

The correct inequalities satisfied by \( a, b, c, d \) are:
\[ b < d < a < c \]