The local maximum value of the function \(f(x) = x^{3} + 3x^{2} - 9x + 1\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 28
The critical points are where \(f'(x) = 0\):
\( f'(x) = 3x^{2} + 6x - 9 \)
\( = 3(x + 3)(x - 1) \)
So x = −3 or x = 1. By the second derivative test, with \(f''(x) = 6x + 6\):
\( f''(-3) = -12 \lt 0 \), so x = −3 is a point of local maximum
\( f''(1) = 12 \gt 0 \), so x = 1 is a point of local minimum
The local maximum value is
\( f(-3) = -27 + 27 + 27 + 1 \)
\( = 28 \)
Why the other options are incorrect
Option (1) \(-4\) — Incorrect. This is f(1) = 1 + 3 − 9 + 1, the local minimum value.
Option (2) 1 — Incorrect. This is f(0); x = 0 is not a critical point.
Option (4) 32 — Incorrect. This is the difference 28 − (−4) of the two local values, not a value of f at a critical point.