The sum of the absolute maximum and minimum values of the function f(x)=|x²-5 x+6|-3 x+2 in the interval [-1,3] is equal to :

Solution

Maths sir · 4 steps

Split the modulus at the roots 2 and 3, then check each piece's ends; both pieces only fall.

1. Remove the modulus. x² - 5x + 6 = (x-2)(x-3). It is non-negative on [-1, 2] and negative on (2, 3), so the function has two different formulas on the interval.
2. Write both pieces. On [-1, 2]: f(x) = x² - 5x + 6 - 3x + 2 = x² - 8x + 8. On [2, 3]: f(x) = -(x² - 5x + 6) - 3x + 2 = -x² + 2x - 4.
3. Check monotonicity. For the first piece f'(x) = 2x - 8 < 0 on [-1, 2]. For the second piece f'(x) = -2x + 2 < 0 on [2, 3]. So f is decreasing on the whole interval, and only the points x = -1, 2, 3 need checking.

   x	Formula used	f(x)
   -1	1 + 8 + 8	17 (max)
   2	4 - 16 + 8	-4
   3	-9 + 6 - 4	-7 (min)

4. So. The absolute maximum is 17 and the absolute minimum is -7, so their sum is 17 + (-7) = 10, which is option (c).

The idea to keep. With a modulus inside, split the interval at the zeros of the expression. Then compare values at the endpoints, the split points and any critical points inside each piece.

Answer from the NTA final answer key. Solution by PrepWiser.