Consider the function \(f(x) = \frac{x^{3}}{3} + \frac{7}{2}x^{2} + 10x + \frac{133}{2}\), \(x \in [-8, 0]\). Which of the following statements is/are correct?
Answer
(C), (D)
Explanation
On a closed interval the extreme values occur at critical points or at the endpoints, so compare all of them.
\(f'(x) = x^{2} + 7x + 10 = (x + 2)(x + 5)\), so there is a local maximum at \(x = -5\) and a local minimum at \(x = -2\). The values are \(f(-8) = \frac{239}{6} \approx 39.83\), \(f(-5) = \frac{187}{3} \approx 62.33\), \(f(-2) = \frac{347}{6} \approx 57.83\) and \(f(0) = \frac{133}{2} = 66.5\).
- The maximum is \(\frac{133}{2}\) at the endpoint \(x = 0\), not at \(x = -5\): C holds and A fails.
- The minimum is at the endpoint \(x = -8\), not at \(x = -2\): B fails.
- \(f'(x) = x^{2} + 7x + 10\) is an upward parabola with its vertex at \(x = -\frac{7}{2}\), which lies inside \([-8, 0]\): D holds.
Answer: C, D