The sum of the absolute maximum and minimum values of the function `f(x) = |x^(2) – 5x + 6| – 3x + 2` in the interval `[–1, 3]` is equal to

Options

A. `12`
B. `10`
C. `24`
D. `13`

Answer

To find the sum of the absolute maximum and minimum values of the function \( f(x) = |x^2 - 5x + 6| - 3x + 2 \) in the interval \([-1, 3]\), we will follow these steps:

### Step 1: Analyze the function inside the absolute value
First, we need to analyze the expression inside the absolute value:
\[ g(x) = x^2 - 5x + 6 \]
This is a quadratic function, and we can find its roots to determine where it changes sign.

### Step 2: Find the roots of \( g(x) \)
To find the roots, we can factor the quadratic:
\[ g(x) = (x - 2)(x - 3) \]
Thus, the roots are \( x = 2 \) and \( x = 3 \). The function \( g(x) \) is positive outside the interval \([2, 3]\) and negative within it.

### Step 3: Determine the intervals for \( f(x) \)
Based on the roots, we can break the function \( f(x) \) into different cases:
- For \( x \in [-1, 2] \): \( g(x) \geq 0 \) so \( f(x) = g(x) - 3x + 2 \)
- For \( x \in [2, 3] \): \( g(x) < 0 \) so \( f(x) = -g(x) - 3x + 2 \)

### Step 4: Calculate \( f(x) \) in each interval
1. **For \( x \in [-1, 2] \)**:
\[ f(x) = (x^2 - 5x + 6) - 3x + 2 = x^2 - 8x + 8 \]

2. **For \( x \in [2, 3] \)**:
\[ f(x) = -(x^2 - 5x + 6) - 3x + 2 = -x^2 + 5x - 6 - 3x + 2 = -x^2 + 2x - 4 \]

### Step 5: Find critical points and evaluate at endpoints
Next, we need to find the critical points of \( f(x) \) in both intervals and evaluate the function at the endpoints.

1. **For \( x \in [-1, 2] \)**:
\[ f'(x) = 2x - 8 \]
Setting \( f'(x) = 0 \):
\[ 2x - 8 = 0 \implies x = 4 \quad (\text{not in } [-1, 2]) \]
Evaluate at endpoints:
- \( f(-1) = (-1)^2 - 8(-1) + 8 = 1 + 8 + 8 = 17 \)
- \( f(2) = 2^2 - 8(2) + 8 = 4 - 16 + 8 = -4 \)

2. **For \( x \in [2, 3] \)**:
\[ f'(x) = -2x + 2 \]
Setting \( f'(x) = 0 \):
\[ -2x + 2 = 0 \implies x = 1 \quad (\text{not in } [2, 3]) \]
Evaluate at endpoints:
- \( f(2) = -4 \) (calculated above)
- \( f(3) = -3^2 + 2(3) - 4 = -9 + 6 - 4 = -7 \)

### Step 6: Determine maximum and minimum values
From the evaluations:
- In \([-1, 2]\): Maximum is \( 17 \) and minimum is \( -4 \).
- In \([2, 3]\): Maximum is \( -4 \) and minimum is \( -7 \).

Thus, the overall maximum value is \( 17 \) and the overall minimum value is \( -7 \).

### Step 7: Calculate the sum of maximum and minimum values
The sum of the absolute maximum and minimum values is:
\[ 17 + (-7) = 10 \]

### Final Answer
The sum of the absolute maximum and minimum values of the function in the interval \([-1, 3]\) is \( \boxed{10} \).