The points of local maxima and local minima of the \( f(x)=x^{3}-6x^{2}+9x+27 \) are
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → local maxima at x = 1 & local minima at x = 3
The critical points are where \( f'(x)=0 \):
\( f'(x)=3x^{2}-12x+9 \)
\( =3(x-1)(x-3) \)
So x = 1 or x = 3. By the second derivative test,
\( f''(x)=6x-12 \)
\( f''(1)=-6\lt0 \)
\( f''(3)=6\gt0 \)
So f has a local maximum at x = 1 and a local minimum at x = 3.