Find sum of maximum and minimum values of the function `f(x) = sin^2x + 8cosx - 7`

Options

A. `-4`
B. `-5`
C. 4
D. 5

Explanation

To find the sum of the maximum and minimum values of the function \( f(x) = \sin^2 x + 8 \cos x - 7 \), we will follow these steps:

### Step 1: Rewrite the function
We know that \( \sin^2 x = 1 - \cos^2 x \). Therefore, we can rewrite the function as:
\[ f(x) = (1 - \cos^2 x) + 8 \cos x - 7 \]
This simplifies to:
\[ f(x) = -\cos^2 x + 8 \cos x - 6 \]

### Step 2: Rearrange the function
Next, we can rearrange the function:
\[ f(x) = -(\cos^2 x - 8 \cos x + 6) \]

### Step 3: Complete the square
To complete the square for the quadratic expression \( \cos^2 x - 8 \cos x \), we can rewrite it as:
\[ \cos^2 x - 8 \cos x = (\cos x - 4)^2 - 16 \]
Thus, substituting back into the function gives:
\[ f(x) = -((\cos x - 4)^2 - 16) - 6 \]
This simplifies to:
\[ f(x) = -(\cos x - 4)^2 + 16 - 6 \]
So we have:
\[ f(x) = -(\cos x - 4)^2 + 10 \]

### Step 4: Determine the range of \( \cos x \)
The range of \( \cos x \) is from -1 to 1. Therefore, the expression \( \cos x - 4 \) will range from:
\[ -1 - 4 = -5 \quad \text{to} \quad 1 - 4 = -3 \]

### Step 5: Find the square of the range
Now, we need to find the square of this range:
\[ (\cos x - 4)^2 \text{ will range from } (-5)^2 = 25 \text{ to } (-3)^2 = 9 \]

### Step 6: Determine the range of \( f(x) \)
Since \( f(x) = -(\cos x - 4)^2 + 10 \), we can find the maximum and minimum values:
- The maximum value occurs when \( (\cos x - 4)^2 \) is at its minimum (which is 9):
\[ f_{\text{max}} = -9 + 10 = 1 \]
- The minimum value occurs when \( (\cos x - 4)^2 \) is at its maximum (which is 25):
\[ f_{\text{min}} = -25 + 10 = -15 \]

### Step 7: Calculate the sum of maximum and minimum values
Now, we can find the sum of the maximum and minimum values:
\[ \text{Sum} = f_{\text{max}} + f_{\text{min}} = 1 + (-15) = -14 \]

### Final Answer
Thus, the sum of the maximum and minimum values of the function \( f(x) \) is:
\[ \boxed{-14} \]