For the function \( f(x)=2x^{3}-21x^{2}+36x-5 \), which of the following statements are correct?
A. \( x=2 \) is a point of local maxima.
B. \( x=6 \) is a point of local minima.
C. The local maximum value is 12.
D. The local maximum value is −94.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → B and C only
\( f'(x)=6x^{2}-42x+36 \)
\( =6(x-1)(x-6) \)
The critical points are x = 1 and x = 6. By the second derivative test:
\( f''(x)=12x-42 \)
\( f''(1)=-30\lt 0 \), so x = 1 is a local maximum
\( f''(6)=30\gt 0 \), so x = 6 is a local minimum
\( f(1)=2-21+36-5=12 \)
\( f(6)=432-756+216-5=-113 \)
So the local maximum is at x = 1 with value 12, and the local minimum is at x = 6 with value −113.
Checking the statements
(A) \( x=2 \) is a point of local maxima. — Incorrect. x = 2 is not a critical point: f′(2) = 6(1)(−4) = −24 ≠ 0.
(B) \( x=6 \) is a point of local minima. — Correct. f″(6) = 30 > 0.
(C) The local maximum value is 12. — Correct. f(1) = 12.
(D) The local maximum value is −94. — Incorrect. The local maximum value is f(1) = 12 (and the local minimum value is −113).