The maximum value of the function \( f(x)=2x^{3}-24x+5 \) in the interval \( [0,4] \) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 37
\( \displaystyle f'(x)=6x^{2}-24 \)
\( \displaystyle =6(x-2)(x+2) \)
Only x = 2 lies in [0, 4]. On a closed interval, the absolute maximum is the largest of the values at the critical point and the end points:
\( \displaystyle f(0)=5 \)
\( \displaystyle f(2)=16-48+5=-27 \)
\( \displaystyle f(4)=128-96+5=37 \)
The maximum value is 37.
Why the other options are incorrect
Option (1) 27 — Incorrect. The values of f at 0, 2 and 4 are 5, −27 and 37; none is 27.
Option (2) 5 — Incorrect. This is f(0), the value at the left end point, not the largest value.
Option (3) −27 — Incorrect. This is f(2), the value at the critical point; it is the minimum on [0, 4], not the maximum.