Question

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

• A. 12
• B. 10
• C. 24
• D. 13

Hint:

When working with absolute value functions, carefully split the function into intervals based on where the absolute value changes. Evaluate critical points and boundaries to find extrema.

Answer 1

The Correct Option is B

To solve the problem of finding the sum of the absolute maximum and minimum values of the function \( f(x) = \left|x^2 - 5x + 6\right| - 3x + 2 \) in the interval \([-1, 3]\), we need to perform the following steps:

1. Factor and analyze the quadratic expression inside the absolute value:

The function inside the absolute value is \( x^2 - 5x + 6 \), which can be factored as:

\(x^2 - 5x + 6 = (x - 2)(x - 3)\)

1. Determine where \( x^2 - 5x + 6 \) changes sign in the interval \([-1, 3]\).

The factored form \( (x-2)(x-3) \) indicates that the expression is zero at \( x = 2 \) and \( x = 3 \). Between these values, the expression changes signs. To find the values of the quadratic expression at key points:

• For \( x < 2 \), choose \( x = 0: \, (0 - 2)(0 - 3) = 6 \), which is positive.
• For \( x \in (2, 3) \), choose \( x = 2.5: \, (2.5 - 2)(2.5 - 3) = -0.25 \), which is negative.

Thus, the function becomes \( \left|x^2 - 5x + 6\right| \), which behaves as:

• For \( x \leq 2: \text{Identity: } (x - 2)(x - 3) \) is non-negative.
• For \( x > 2: \text{Absolute Value Change: } |(x-2)(x-3)| = -(x-2)(x-3) \).

1. Evaluate \( f(x) \) over these intervals and key points.
2. Calculate \( f(x) \) at crucial points within the interval \([-1, 3]\): \( x = -1, 0, 2, 3 \).

\( f(-1) = |(-1)^2 - 5(-1) + 6| - 3(-1) + 2 = |1 + 5 + 6| + 3 + 2 = 17 \)

\( f(0) = |0^2 - 5(0) + 6| - 3(0) + 2 = 6 + 2 = 8 \)

\( f(2) = |2^2 - 5(2) + 6| - 3(2) + 2 = 0 - 6 + 2 = -4 \)

\( f(3) = |3^2 - 5(3) + 6| - 3(3) + 2 = |0| - 9 + 2 = -7 \)

1. Identify the absolute maximum and minimum values.

From the calculated values:

• The absolute maximum value of \( f(x) \) is 17, occurring at \( x = -1 \).
• The absolute minimum value of \( f(x) \) is -7, occurring at \( x = 3 \).

1. Compute the sum of the absolute maximum and minimum values.

The sum is:

\(17 + (-7) = 10\)

This matches the correct answer, which is:

10