Let \(f(x)=x^3-6x^2+9x-8\) be a function, then which of the following statements are TRUE?
(A) \(f'(x)=3(x-1)(x-3)\)
(B) The critical points of the function are \(x= 1\) and \(x = 3\)
(C) \(x= 1\) is the point of local minimum
(D) The local maximum value is −4
Choose the correct answer from the options given below.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (D) only
Differentiating and factorising:
\(f'(x) = 3x^2 - 12x + 9\)
\(= 3(x^2 - 4x + 3)\)
\(= 3(x - 1)(x - 3)\)
The critical points are where \(f'(x) = 0\):
\(x = 1 \text{ or } x = 3\)
By the second derivative test:
\(f''(x) = 6x - 12\)
\(f''(1) = -6 \lt 0\), so x = 1 is a point of local maximum
\(f''(3) = 6 \gt 0\), so x = 3 is a point of local minimum
\(f(1) = 1 - 6 + 9 - 8 = -4\)
So the critical points are x = 1 and x = 3, x = 1 is the point of local maximum, and the local maximum value is \(f(1) = -4\).
Checking the statements
(A) \(f'(x)=3(x-1)(x-3)\) — Correct. \(3x^2 - 12x + 9 = 3(x - 1)(x - 3)\).
(B) The critical points of the function are \(x= 1\) and \(x = 3\) — Correct. \(f'(x) = 0\) at these two points.
(C) \(x= 1\) is the point of local minimum — Incorrect. \(f''(1) = -6 \lt 0\), so x = 1 is a local maximum.
(D) The local maximum value is −4 — Correct. \(f(1) = -4\).