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

Given the function \( f(x) = x^2 - 5x + 6 - 3x + 2 = x^2 - 8x + 8 \). We want to find the absolute maximum and minimum values of \( f(x) \) on the interval \([-1, 3]\).
First, we find the critical points by taking the derivative and setting it to zero:
\[f'(x) = 2x - 8\]
\[f'(x) = 0 \implies 2x - 8 = 0 \implies 2x = 8 \implies x = 4\]
Since \( x = 4 \) is outside the interval \([-1, 3]\), there are no critical points within the interval.
Now we evaluate \( f(x) \) at the endpoints of the interval:
\[\text{For } x = -1:\]
\[f(-1) = (-1)^2 - 8(-1) + 8 = 1 + 8 + 8 = 17\]
\[\text{For } x = 3:\]
\[f(3) = 3^2 - 8(3) + 8 = 9 - 24 + 8 = -7\]
The absolute maximum value is \( 17 \) (at \( x = -1 \)), and the absolute minimum value is \( -7 \) (at \( x = 3 \)).
The sum of the absolute maximum and minimum values is:
\[17 + (-7) = 10.\]