Determine any relative extrema. Select the correct choice below​ and, if​ necessary, fill in the answer​ box(es) to complete your choice.
A.
The function has a relative maximum at enter your response here
​, 
but does not have any relative minima.
​(Use a comma to separate answers as needed. Type an ordered​ pair.)
B.
The function has a relative maximum at left parenthesis negative 3 StartRoot 2 EndRoot comma f left parenthesis negative 3 StartRoot 2 EndRoot right parenthesis right parenthesis
 
and a relative minimum at left parenthesis 3 StartRoot EndRoot
.
​(Use a comma to separate answers as needed. Type ordered​ pairs.)
C.
The function has a relative minimum at enter your response here
​, 
but does not have any relative maxima.
​(Use a comma to separate answers as needed. Type an ordered​ pair.)
D.
The function does not have any relative extrema.


To determine the relative extrema of a function, we need to find the critical points by setting the derivative equal to zero and checking for sign changes in the derivative around these points. Since the specific function isn't provided, let's outline the steps to analyze it:

1. **Find the Derivative**: Differentiate the function.
2. **Set the Derivative to Zero**: Solve for \( x \) when \( f'(x) = 0 \).
3. **Determine Critical Points**: Identify critical points from step 2.
4. **Second Derivative Test or First Derivative Test**: Use either test to classify each critical point as a relative maximum, minimum, or neither.

### Answer Options

Assuming we followed these steps and found critical points, we would fill in the answer as follows:

**If we found:**
- A relative maximum at \( (-3\sqrt{2}, f(-3\sqrt{2})) \)
- A relative minimum at \( (3\sqrt{2}, f(3\sqrt{2})) \)

Then the correct choice would be:

**B.**
The function has a relative maximum at \( (-3\sqrt{2}, f(-3\sqrt{2})) \) and a relative minimum at \( (3\sqrt{2}, f(3\sqrt{2})) \).

### Next Steps

If you provide the specific function, I can assist you in performing these calculations to identify the exact points of relative extrema. 

Do you want to go through the steps in detail, or do you have a specific function in mind? Here are some follow-up questions:

1. What is the specific function you are analyzing?
2. Would you like a detailed explanation of the derivative tests?
3. Are you interested in understanding how to sketch the function based on its extrema?
4. Do you need help with any specific type of function (polynomial, trigonometric, etc.)?
5. Would you like to see examples of relative extrema for different functions? 

**Tip:** Always check the endpoints of the function's domain, as they can also provide information about absolute extrema, even if they are not relative extrema.

Determine any relative extrema. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
A. The function has a relative maximum at enter your response here, but does not have any relative minima.
(Use a comma to separate answers as needed. Type an ordered pair.)
B. The function has a relative maximum at (-3√2, f(-3√2)) and a relative minimum at (3√2, f(3√2)).
(Use a comma to separate answers as needed. Type ordered pairs.)
C. The function has a relative minimum at enter your response here, but does not have any relative maxima.
(Use a comma to separate answers as needed. Type an ordered pair.)
D. The function does not have any relative extrema.

Learn how to determine the relative extrema of a function using derivative tests. This example includes solving for critical points, testing for maximums and minimums, and understanding function behavior around extrema points. Suitable for high school calculus students.

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