Let \( f(x) = (x + 3)^2 (x - 2)^3 \), \( x \in [-4, 4] \). If \( M \) and \( m \) are the maximum and minimum values of \( f \), respectively in \([-4, 4]\), then the value of \( M - m \) is:
- 600
- 392
- 608
- 108
The Correct Option is C
Solution and Explanation
To find the maximum and minimum values of \( f(x) \):
Take the derivative \( f'(x) \) and find the critical points.
Evaluate \( f(x) \) at critical points and endpoints \( x = -4, -3, -2, -1, 1, 2, 3, 4 \).
The maximum value \( M = 392 \) and the minimum value \( m = -216 \).
The value of \( M - m \) is:
\[ M - m = 392 - (-216) = 608. \]
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