The sum of the absolute maximum value and the absolute minimum value of \(f(x) = 3x^4 - 8x^3 + 6\) on the interval [−1, 3] is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 23
The critical points are where f’(x) = 0:
\(f'(x) = 12x^3 - 24x^2\)
\(f'(x) = 12x^2(x - 2)\)
So x = 0 or x = 2, both inside [−1, 3]. Comparing the values at the critical points and the end points:
\(f(-1) = 3 + 8 + 6 = 17\)
\(f(0) = 6\)
\(f(2) = 48 - 64 + 6 = -10\)
\(f(3) = 243 - 216 + 6 = 33\)
The absolute maximum is 33 and the absolute minimum is −10. Their sum:
\(33 + (-10) = 23\)
Why the other options are incorrect
Option (1) 7 — Incorrect. This is 17 + (−10), taking f(−1) = 17 as the maximum and missing the end point x = 3.
Option (3) 43 — Incorrect. This is 33 − (−10), the difference of the two values, not their sum.
Option (4) 39 — Incorrect. This is 33 + 6, taking f(0) = 6 as the minimum; f(2) = −10 is smaller.