For the function \(f(x) = (x - 1)^{2} + 3\) on the interval [−3, 2]:
(A) The only critical point of f in (−3, 2) is x = 1.
(B) The absolute minimum value of f is 3.
(C) The absolute maximum value of f is 19.
(D) f attains its absolute maximum at x = 2.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (B) and (C) only
Differentiating,
\(f'(x) = 2(x - 1)\), which is 0 only at x = 1
On a closed interval the absolute extrema occur at critical points or at end points:
\( f(-3) = 16 + 3 = 19 \)
\( f(1) = 0 + 3 = 3 \)
\( f(2) = 1 + 3 = 4 \)
So the only critical point is x = 1, the absolute minimum is 3 (at x = 1), and the absolute maximum is 19 (at x = −3).
Checking the statements
(A) The only critical point of f in (−3, 2) is x = 1. — Correct. f′(x) = 2(x − 1) vanishes only at x = 1.
(B) The absolute minimum value of f is 3. — Correct. f(1) = 3 is the least of 19, 3 and 4.
(C) The absolute maximum value of f is 19. — Correct. f(−3) = 19 is the greatest of the three values.
(D) f attains its absolute maximum at x = 2. — Incorrect. f(2) = 4, while f(−3) = 19; the maximum is at the other end point.