A function \( f(x) \) is defined as:
\( f(x)=x^{3}-6x^{2}+9x+15 \)
For the function \( f(x) \), which of the following statements are correct ?
(A) The critical points are \( x=1 \) and \( x=3 \).
(B) \( x=1 \) is a point of local minimum.
(C) The local maximum value is 19.
(D) \( x=3 \) is a point of local minimum.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (C), and (D) only
The critical points are where \( f'(x)=0 \), and the sign of \( f''(x) \) at each tells whether it is a maximum or a minimum:
\( \displaystyle f'(x)=3x^{2}-12x+9 \)
\( \displaystyle =3(x-1)(x-3) \)
\( \displaystyle f''(x)=6x-12 \)
\( \displaystyle f''(1)=-6\lt 0 \)
\( \displaystyle f''(3)=6\gt 0 \)
\( \displaystyle f(1)=1-6+9+15=19 \)
So the critical points are x = 1 and x = 3, x = 1 is a point of local maximum with local maximum value 19, and x = 3 is a point of local minimum.
Checking the statements
(A) The critical points are \( x=1 \) and \( x=3 \). — Correct. \( f'(x)=3(x-1)(x-3)=0 \) gives x = 1 and x = 3.
(B) \( x=1 \) is a point of local minimum. — Incorrect. \( f''(1)=-6\lt 0 \), so x = 1 is a point of local maximum.
(C) The local maximum value is 19. — Correct. \( f(1)=19 \).
(D) \( x=3 \) is a point of local minimum. — Correct. \( f''(3)=6\gt 0 \).
NCERT (2026-27 edition):
Maxima and Minima
