The local maximum value of the function \(f(x) = x^3 - 27x + 10\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 64
The critical points are where f′(x) = 0.
\(f'(x) = 3x^2 - 27\)
\(f'(x) = 3(x - 3)(x + 3)\)
So x = 3 or x = −3. By the second derivative test:
\(f''(x) = 6x\)
\(f''(-3) = -18 \lt 0\), local maximum
\(f''(3) = 18 \gt 0\), local minimum
The local maximum value is
\(f(-3) = -27 + 81 + 10 = 64\)
Why the other options are incorrect
Option (1) −44 — Incorrect. This is the local minimum value, f(3) = 27 − 81 + 10.
Option (3) −3 — Incorrect. This is the point x where the maximum occurs, not the maximum value.
Option (4) 3 — Incorrect. x = 3 is the point of local minimum, not a value of f.