For the function \( f(x)=2x^{3}-3x^{2}-12x+5 \). Which of the following statements is/are TRUE?
(A) Function \( f(x) \) has two points of local maximum
(B) Function \( f(x) \) has two points of local minimum
(C) Function \( f(x) \) has one local maxima and one local minima
(D) Function \( f(x) \) has local maxima at \( x=-1 \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (C) and (D) only
The critical points come from \( f'(x)=0 \):
\( \displaystyle f'(x)=6x^{2}-6x-12 \)
\( \displaystyle =6(x-2)(x+1) \)
So x = −1 and x = 2. By the second derivative test,
\( \displaystyle f''(x)=12x-6 \)
\( \displaystyle f''(-1)=-18\lt 0 \)
\( \displaystyle f''(2)=18\gt 0 \)
So f has exactly one local maximum, at x = −1, and exactly one local minimum, at x = 2.
Checking the statements
(A) Function \( f(x) \) has two points of local maximum — Incorrect. The only local maximum is at x = −1.
(B) Function \( f(x) \) has two points of local minimum — Incorrect. The only local minimum is at x = 2.
(C) Function \( f(x) \) has one local maxima and one local minima — Correct. Maximum at x = −1, minimum at x = 2.
(D) Function \( f(x) \) has local maxima at \( x=-1 \) — Correct. \( f''(-1)=-18\lt 0 \).