The absolute maximum value of \(f(x) = x^2 - 4x + 1\) on the interval [0, 5] is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 6
On a closed interval the absolute maximum is the largest of the values at the critical points and at the end points.
\(f'(x) = 2x - 4\)
\(2x - 4 = 0\)
\(x = 2\)
The values to compare:
\(f(0) = 1\)
\(f(2) = 4 - 8 + 1 = -3\)
\(f(5) = 25 - 20 + 1 = 6\)
The largest is 6, at the end point x = 5. (The critical point x = 2 gives the absolute minimum, −3.)
Why the other options are incorrect
Option (1) \(-3\) — Incorrect. This is f(2), the absolute minimum value.
Option (3) 1 — Incorrect. This is f(0); f(5) = 6 is larger.
Option (4) 2 — Incorrect. This is the critical point x = 2, not a value of f.