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…

Given: $f(x) = x^3 - 6x^2 + 9x - 8$f(x)=x3−6x2+9x−8

The first derivative is:

$$f'(x) = 3x^2 - 12x + 9$$f′(x)=3x2−12x+9

$$f'(x) = 3(x^2 - 4x + 3)$$f′(x)=3(x2−4x+3)

$$f'(x) = 3(x - 1)(x - 3)$$f′(x)=3(x−1)(x−3)

Statement (A) is TRUE.

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For critical points, set $f'(x) = 0$f′(x)=0:

$$3(x - 1)(x - 3) = 0$$3(x−1)(x−3)=0

$$(x - 1)(x - 3) = 0$$(x−1)(x−3)=0

The critical points are $x = 1$x=1 and $x = 3$x=3.

Statement (B) is TRUE.

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The second derivative is:

$$f''(x) = 6x - 12$$f′′(x)=6x−12

At $x = 1$x=1:

$$f''(1) = 6(1) - 12$$f′′(1)=6(1)−12

$$f''(1) = -6$$f′′(1)=−6

Since $f''(1) < 0$f′′(1)<0, the point $x = 1$x=1 is a local maximum.

Statement (C) is FALSE.

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The local maximum value at $x = 1$x=1:

$$f(1) = (1)^3 - 6(1)^2 + 9(1) - 8$$f(1)=(1)3−6(1)2+9(1)−8

$$f(1) = 1 - 6 + 9 - 8$$f(1)=1−6+9−8

$$f(1) = -4$$f(1)=−4

Statement (D) is TRUE.

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The true statements are (A), (B), and (D).