Let the maximum and minimum value of the expression `2 cos^(2) theta + cos theta + 1` is M and m respectively, then the value of `[M/m]` is (where [.] is the greatest integer function)

Answer

To find the maximum and minimum values of the expression \( f(\theta) = 2 \cos^2 \theta + \cos \theta + 1 \), we will follow these steps:

### Step 1: Rewrite the expression
Let \( f(\theta) = 2 \cos^2 \theta + \cos \theta + 1 \).

### Step 2: Differentiate the function
To find the critical points, we differentiate \( f(\theta) \) with respect to \( \theta \):
\[ f'(\theta) = \frac{d}{d\theta}(2 \cos^2 \theta + \cos \theta + 1) = -4 \cos \theta \sin \theta - \sin \theta \]
This simplifies to:
\[ f'(\theta) = -\sin \theta (4 \cos \theta + 1) \]

### Step 3: Set the derivative to zero
Setting \( f'(\theta) = 0 \) gives us:
\[ -\sin \theta (4 \cos \theta + 1) = 0 \]
This leads to two cases:
1. \( \sin \theta = 0 \)
2. \( 4 \cos \theta + 1 = 0 \) or \( \cos \theta = -\frac{1}{4} \)

### Step 4: Solve for critical points
1. From \( \sin \theta = 0 \):
- \( \theta = n\pi \) for \( n \in \mathbb{Z} \)
- \( \cos \theta = 1 \) or \( \cos \theta = -1 \)

2. From \( \cos \theta = -\frac{1}{4} \):
- This gives us another critical point.

### Step 5: Evaluate \( f(\theta) \) at critical points
1. When \( \cos \theta = 1 \):
\[ f(\theta) = 2(1)^2 + 1 + 1 = 4 \]

2. When \( \cos \theta = -1 \):
\[ f(\theta) = 2(-1)^2 + (-1) + 1 = 2 \]

3. When \( \cos \theta = -\frac{1}{4} \):
\[ f(\theta) = 2\left(-\frac{1}{4}\right)^2 + \left(-\frac{1}{4}\right) + 1 \]
\[ = 2\left(\frac{1}{16}\right) - \frac{1}{4} + 1 = \frac{1}{8} - \frac{2}{8} + \frac{8}{8} = \frac{7}{8} \]

### Step 6: Determine maximum and minimum values
From the evaluations:
- Maximum value \( M = 4 \)
- Minimum value \( m = \frac{7}{8} \)

### Step 7: Calculate \( \frac{M}{m} \)
\[ \frac{M}{m} = \frac{4}{\frac{7}{8}} = 4 \times \frac{8}{7} = \frac{32}{7} \]

### Step 8: Apply the greatest integer function
Now we apply the greatest integer function:
\[ \left\lfloor \frac{32}{7} \right\rfloor = \left\lfloor 4.5714 \right\rfloor = 4 \]

### Final Answer
Thus, the value of \( \left\lfloor \frac{M}{m} \right\rfloor \) is \( 4 \).

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