For the function \( f(x)=2x^{3}-21x^{2}+36x-20 \), which of the following statements are correct?
(A). \( x=6 \) is a point of local minimum.
(B). \( x=2 \) is a point of local maximum.
(C). The local minimum value is -105.
(D). The local maximum value is -3.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (D) only
Differentiating,
\( f'(x)=6x^{2}-42x+36 \)
\( =6(x-1)(x-6) \)
So the critical points are \( x=1 \) and \( x=6 \). The second derivative is
\( f''(x)=12x-42 \)
At \( x=1 \), \( f''(1)=-30\lt 0 \), a local maximum:
\( f(1)=2-21+36-20=-3 \)
At \( x=6 \), \( f''(6)=30\gt 0 \), a local minimum:
\( f(6)=432-756+216-20=-128 \)
So the local maximum is at \( x=1 \) with value \( -3 \), and the local minimum is at \( x=6 \) with value \( -128 \).
Checking the statements
(A) \( x=6 \) is a point of local minimum. — Correct. \( f''(6)=30\gt 0 \).
(B) \( x=2 \) is a point of local maximum. — Incorrect. \( f'(2)=-24\ne 0 \), so \( x=2 \) is not even a critical point; the maximum is at \( x=1 \).
(C) The local minimum value is -105. — Incorrect. The local minimum value is \( f(6)=-128 \).
(D) The local maximum value is -3. — Correct. \( f(1)=-3 \).