Question:

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?

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For finding maxima and minima, use the first derivative to identify critical points. To determine whether these points represent maxima or minima, you can use the second derivative or analyze the function's behavior at the endpoints of the interval.

Updated On: Apr 4, 2025

The maximum value of \( f \) is attained at \( x = -5 \)
The minimum value of \( f \) is attained at \( x = -2 \)
The maximum value of \( f \) is \( \frac{133}{2} \)
The minimum value of the derivative of \( f \) is attained at \( x = -\frac{7}{2} \)

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The Correct Option is C, D

Solution and Explanation

We calculate the first derivative of the function \( f(x) = \frac{x^3}{3} + \frac{7}{2}x^2 + 10x + \frac{133}{2} \), which gives:

\[ f'(x) = x^2 + 7x + 10 \] Solving \( f'(x) = 0 \), we find the critical points \( x = -2 \) and \( x = -5 \).

Evaluating the function at the critical points and endpoints:
\( f(-2) = \frac{347}{6} \)
\( f(-5) = \frac{187}{3} \)
\( f(-8) = \frac{239}{6} \)
\( f(0) = \frac{133}{2} \)

The maximum value is \( \frac{133}{2} \) at \( x = 0 \), and the minimum value occurs at \( x = -8 \).

Additionally, the minimum of the derivative occurs at \( x = -\frac{7}{2} \).

Thus, the correct answers are Option (C) and Option (D).

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