The function \( f(x)=-x^3+12x^2-36x+21 \) is
A. Increasing on \( (-\infty,2) \)
B. Decreasing on \( (-\infty,2) \)
C. Increasing on \( (2,6) \)
D. Decreasing on \( (6,\infty) \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → B, C and D only
A function increases where f'(x) > 0 and decreases where f'(x) < 0, so the sign of the derivative decides each statement.
\( \displaystyle f'(x)=-3x^2+24x-36 \)
\( \displaystyle =-3(x^2-8x+12) \)
\( \displaystyle =-3(x-2)(x-6) \)
The product (x − 2)(x − 6) is positive outside [2, 6] and negative between 2 and 6. Because of the factor −3, the sign of f'(x) is the opposite:
\( \displaystyle x\lt 2:\ f'(x)\lt 0 \)
\( \displaystyle 2\lt x\lt 6:\ f'(x)\gt 0 \)
\( \displaystyle x\gt 6:\ f'(x)\lt 0 \)
So f is decreasing on \( (-\infty,2) \), increasing on \( (2,6) \) and decreasing on \( (6,\infty) \).
Checking the statements
(A) Increasing on \( (-\infty,2) \) — Incorrect. f'(x) < 0 there, so f is decreasing.
(B) Decreasing on \( (-\infty,2) \) — Correct. f'(x) < 0 for x < 2.
(C) Increasing on \( (2,6) \) — Correct. f'(x) > 0 for 2 < x < 6.
(D) Decreasing on \( (6,\infty) \) — Correct. f'(x) < 0 for x > 6.