Let \( f(x) \) be a function defined by \( f(x)=2x^3-24x+107 \). If α and β are the maximum and minimum values respectively in the interval [1, 3], then α − β is equal to:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 14
On a closed interval, the absolute maximum and minimum occur at a critical point inside the interval or at an end point.
\( \displaystyle f'(x)=6x^2-24 \)
\( \displaystyle 6x^2-24=0\ \Rightarrow\ x=\pm 2 \)
Only x = 2 lies in [1, 3]. Values:
\( \displaystyle f(1)=2-24+107=85 \)
\( \displaystyle f(2)=16-48+107=75 \)
\( \displaystyle f(3)=54-72+107=89 \)
So α = 89 (at the end point x = 3) and β = 75, and
\( \displaystyle \alpha-\beta=89-75=14 \)