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
Given: $f(x)=x^3-6x^2+9x-8$
$f'(x)=3x^2-12x+9=3(x^2-4x+3)=3(x-1)(x-3)$ → (A) True
Critical points from $f'(x)=0$ ⇒ $x=1,3$ → (B) True
$f''(x)=6x-12$
At $x=1$: $f''(1)=-6<0$ ⇒ local maximum; $f(1)=-4$ → (C) False, (D) True
At $x=3$: $f''(3)=6>0$ ⇒ local minimum; $f(3)=-8$
Correct statements: (A), (B), (D)