Questions: Determine the location of each local extremum of the function. f(x)=-x^3-9x^2-24x+1 What is/are the local minimum/minima? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

Determine the location of each local extremum of the function. f(x)=-x^3-9x^2-24x+1 What is/are the local minimum/minima? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
Transcript text: Determine the location of each local extremum of the function. \[ f(x)=-x^{3}-9 x^{2}-24 x+1 \] What is/are the local minimum/minima? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
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Solution

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Solution Steps

To find the local extrema of the function \( f(x) = -x^3 - 9x^2 - 24x + 1 \), we need to follow these steps:

  1. Compute the first derivative of the function, \( f'(x) \).
  2. Find the critical points by setting \( f'(x) = 0 \) and solving for \( x \).
  3. Use the second derivative test to determine the nature of each critical point (local minimum or maximum).
Step 1: Find the First Derivative

The first derivative of the function \( f(x) = -x^3 - 9x^2 - 24x + 1 \) is calculated as follows: \[ f'(x) = -3x^2 - 18x - 24 \]

Step 2: Determine Critical Points

To find the critical points, we set the first derivative equal to zero: \[ -3x^2 - 18x - 24 = 0 \] Factoring or using the quadratic formula, we find the critical points: \[ x = -4, \quad x = -2 \]

Step 3: Find the Second Derivative

Next, we compute the second derivative to determine the nature of the critical points: \[ f''(x) = -6x - 18 \]

Step 4: Classify the Critical Points

We evaluate the second derivative at each critical point:

  • For \( x = -4 \): \[ f''(-4) = -6(-4) - 18 = 24 - 18 = 6 \quad (\text{local minimum}) \]
  • For \( x = -2 \): \[ f''(-2) = -6(-2) - 18 = 12 - 18 = -6 \quad (\text{local maximum}) \]

Final Answer

The local minimum occurs at \( x = -4 \) and the local maximum occurs at \( x = -2 \). Thus, the final answer is: \[ \boxed{x = -4} \quad \text{(local minimum)} \]

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