Given: f(x)=x3−6x2+9x−8
The first derivative is:
f′(x)=3x2−12x+9
f′(x)=3(x2−4x+3)
f′(x)=3(x−1)(x−3)
Statement (A) is TRUE.
For critical points, set f′(x)=0:
3(x−1)(x−3)=0
(x−1)(x−3)=0
The critical points are x=1 and x=3.
Statement (B) is TRUE.
The second derivative is:
f′′(x)=6x−12
At x=1:
f′′(1)=6(1)−12
f′′(1)=−6
Since f′′(1)<0, the point x=1 is a local maximum.
Statement (C) is FALSE.
The local maximum value at x=1:
f(1)=(1)3−6(1)2+9(1)−8
f(1)=1−6+9−8
f(1)=−4
Statement (D) is TRUE.
The true statements are (A), (B), and (D).