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

Answer 1

Answer: (B)

10

Answer 2

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f(x) = x2-5x+6-3x+2
f'(x) = 2x-5-3 = 2x-8
f'(x) = 0
2x-8 = 0
2x = 8
x = 4

⇒Since x=4 is outside the interval [−1,3], there are no critical points within the interval.

Evaluate f(x) at the endpoints of the interval:
For x = -1
f(−1)=(−1)2−5(−1)+6−3(−1)+2 = 1+5+6+3+2 = 17
For x = 3
f(3)=32−5(3)+6−3(3)+2 = 9−15+6−9+2 = −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

So, the correct option is (B): 10