Let f be a function defined by $f(x) = 2x^3-3x^2-36x+2$, then which of the following are correct?
(A) The critical points of f(x) are -2 and 3.
(B) The function f(x) increases in the interval (3, ∞)
(C) The function f(x) decreases in the interval (-2, 3)
(D) The function f(x) increases in the interval (-2, 3)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (C) only **
Given function:
$f(x)=2x^{3}-3x^{2}-36x+2$
Derivative:
$f'(x)=6(x-3)(x+2)$
Critical points:
Roots of $f'(x)$ are $x=-2$ and $x=3$.
Sign pattern of $f'(x)$:
• On $(-\infty,-2)$, $f'(x)>0$ (increasing)
• On $(-2,3)$, $f'(x)<0$ (decreasing)
• On $(3,\infty)$, $f'(x)>0$ (increasing)
Conclusion:
(A) True — critical points are $-2$ and $3$
(B) True — $f$ increases on $(3,\infty)$
(C) True — $f$ decreases on $(-2,3)$
(D) False — it does not increase on $(-2,3)$
Correct statements: A, B, C