For the function \( f(x)=\displaystyle\frac{x^4}{4}-2x^3+\displaystyle\frac{11}{2}x^2-6x \), which of the following statements are TRUE?
(A) Critical points of \( f(x) \) are \( x=1 \), 2 and 3.
(B) \( f(x) \) is increasing over (1,2).
(C) \( f(x) \) is decreasing over (2,3).
(D) \( f(x) \) is increasing over \( (3,\infty) \).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (B), (C) and (D)
Differentiating,
\( \displaystyle f'(x)=x^3-6x^2+11x-6 \)
x = 1 makes this zero (1 − 6 + 11 − 6 = 0), so \( (x-1) \) is a factor:
\( \displaystyle f'(x)=(x-1)(x^2-5x+6) \)
\( \displaystyle =(x-1)(x-2)(x-3) \)
Sign of \( f'(x) \) on each interval:
\( \displaystyle 1\lt x\lt 2:\ (+)(-)(-)\gt 0 \)
\( \displaystyle 2\lt x\lt 3:\ (+)(+)(-)\lt 0 \)
\( \displaystyle x\gt 3:\ (+)(+)(+)\gt 0 \)
So the critical points are x = 1, 2 and 3, \( f'(x)\gt 0 \) on (1, 2) and on \( (3,\infty) \), and \( f'(x)\lt 0 \) on (2, 3).
Checking the statements
(A) Critical points of \( f(x) \) are \( x=1 \), 2 and 3. — Correct. \( f'(x)=0 \) exactly at x = 1, 2, 3.
(B) \( f(x) \) is increasing over (1,2). — Correct. \( f'(x)\gt 0 \) on (1, 2).
(C) \( f(x) \) is decreasing over (2,3). — Correct. \( f'(x)\lt 0 \) on (2, 3).
(D) \( f(x) \) is increasing over \( (3,\infty) \). — Correct. \( f'(x)\gt 0 \) for x > 3.