Find `max^m` & `min^(m)` value of `11+cos^(2)alpha`.

Answer

To find the maximum and minimum values of the expression \(11 + \cos^2 \alpha\), we can follow these steps:

### Step 1: Understand the range of \(\cos^2 \alpha\)
The function \(\cos^2 \alpha\) represents the square of the cosine function. The cosine function oscillates between -1 and 1, so when we square it, \(\cos^2 \alpha\) will range from:
\[ \cos^2 \alpha \in [0, 1] \]

### Step 2: Determine the minimum value of \(11 + \cos^2 \alpha\)
The minimum value of \(\cos^2 \alpha\) is 0 (when \(\alpha = \frac{\pi}{2} + k\pi\) for any integer \(k\)). Therefore, we can find the minimum value of the expression:
\[ \text{Minimum value} = 11 + \min(\cos^2 \alpha) = 11 + 0 = 11 \]

### Step 3: Determine the maximum value of \(11 + \cos^2 \alpha\)
The maximum value of \(\cos^2 \alpha\) is 1 (when \(\alpha = 2k\pi\) for any integer \(k\)). Therefore, we can find the maximum value of the expression:
\[ \text{Maximum value} = 11 + \max(\cos^2 \alpha) = 11 + 1 = 12 \]

### Conclusion
Thus, the maximum and minimum values of the expression \(11 + \cos^2 \alpha\) are:
\[ \text{max} = 12 \quad \text{and} \quad \text{min} = 11 \]