https://file.aiquickdraw.com/mathbot/file/1728236622769myj0535e.jpg

Let's solve this problem step-by-step.

We are given the function \( f(x) = \cos(x) - 8x \) over the interval \( [0, 4\pi] \) and are asked to find the relative extrema (minima and maxima) using the Second Derivative Test.

### 1. First Derivative (Critical Points)
To find the critical points, we start by taking the derivative of \( f(x) \).

\[
f'(x) = -\sin(x) - 8
\]

Set the derivative equal to zero to find the critical points:

\[
-\sin(x) - 8 = 0
\]

\[
\sin(x) = -8
\]

However, the sine function only takes values between -1 and 1, so \( \sin(x) = -8 \) has no solutions. This means there are **no critical points** from setting the first derivative equal to zero.

### 2. Endpoints
Since there are no critical points from the derivative, we now evaluate the function at the endpoints of the interval \( [0, 4\pi] \).

- At \( x = 0 \):

\[
f(0) = \cos(0) - 8(0) = 1
\]

- At \( x = 4\pi \):

\[
f(4\pi) = \cos(4\pi) - 8(4\pi) = 1 - 32\pi
\]

### 3. Second Derivative Test
The second derivative is:

\[
f''(x) = -\cos(x)
\]

Since we found no critical points, the second derivative test doesn't directly apply in this situation.

### 4. Conclusion (Relative Extrema)
- **Relative Maximum**: At \( x = 0 \), the function value is \( f(0) = 1 \). This is the highest value in the interval, so the relative maximum occurs at \( (0, 1) \).

- **Relative Minimum**: At \( x = 4\pi \), the function value is \( f(4\pi) = 1 - 32\pi \). This is the lowest value, so the relative minimum occurs at \( (4\pi, 1 - 32\pi) \).

So, the correct answers are:
- Relative minimum: \( (4\pi, 1 - 32\pi) \)
- Relative maximum: \( (0, 1) \)

Would you like any more details, or do you have questions on this? Here are some follow-up questions:

1. How do you handle cases where the sine function yields impossible values like in this example?
2. What happens when the first derivative has no critical points?
3. Can you explain how the Second Derivative Test helps determine concavity?
4. What is the significance of evaluating endpoints in optimization problems?
5. How would this method change if we were working with an open interval?

**Tip**: Always check endpoints in a closed interval when looking for extrema, especially if there are no critical points from the derivative.

Find all relative extrema for f(x) = cos(x) - 8x, on the interval [0, 4π], using the Second Derivative Test.

Step-by-step solution to find the relative extrema of f(x) = cos(x) - 8x on the interval [0, 4π]. This problem demonstrates the use of derivatives and the Second Derivative Test in optimization, along with evaluating endpoints.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation