For the function \( f(x)=2x^3-9x^2+12x-5,x\in[0,3] \), which of the following statements are correct?
A. The point of absolute maxima is \( x=3 \).
B. The absolute minimum value is −1.
C. The critical points are \( x=1 \) and \( x=2 \) .
D. The absolute maximum value is 3.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → A and C only
The critical points are where \( f'(x)=0 \):
\( \displaystyle f'(x)=6x^2-18x+12 \)
\( \displaystyle f'(x)=6(x-1)(x-2) \)
\( \displaystyle f'(x)=0\ \Rightarrow\ x=1,\ x=2 \)
On a closed interval, f is compared at the critical points and at the end points:
\( \displaystyle f(0)=-5 \)
\( \displaystyle f(1)=2-9+12-5=0 \)
\( \displaystyle f(2)=16-36+24-5=-1 \)
\( \displaystyle f(3)=54-81+36-5=4 \)
So the critical points are x = 1 and x = 2, the absolute maximum value is 4 at x = 3, and the absolute minimum value is −5 at x = 0.
Checking the statements
(A) The point of absolute maxima is \( x=3 \). — Correct. f(3) = 4 is the largest value.
(B) The absolute minimum value is −1. — Incorrect. −1 at x = 2 is only a local minimum; the end point gives f(0) = −5.
(C) The critical points are \( x=1 \) and \( x=2 \) . — Correct. These are the zeros of \( f'(x)=6(x-1)(x-2) \).
(D) The absolute maximum value is 3. — Incorrect. The absolute maximum value is 4; 3 is the point where it occurs.
Tip: For absolute extrema on a closed interval, never skip the end points: here both the absolute maximum and the absolute minimum occur at end points.