Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

A.

The function f has an absolute maximum of

enter your response here

at

xequals=enter your response here

and no absolute minimum.

​(Type your answers using exponential notation. Use a comma to separate answers as​ needed.)

B.

The function f has an absolute minimum of

enter your response here

at

xequals=enter your response here

and no absolute maximum.

​(Type your answers using exponential notation. Use a comma to separate answers as​ needed.)

C.

The function f has an absolute minimum of

enter your response here

at

xequals=enter your response here

and an absolute maximum of

enter your response here

at

xequals=enter your response here.

​(Type your answers using exponential notation. Use a comma to separate answers as​ needed.)

D.

There are no local extreme points.

To address this question, we'll need to analyze the function \( f \), which is not explicitly provided in your request. Normally, we would investigate the critical points and behavior of the function to identify its extreme points. Here’s a typical approach to determining which of the options applies:

1. **Find the critical points**: These are where the derivative \( f'(x) \) is zero or undefined. Critical points are potential locations for maxima, minima, or saddle points.
  
2. **Examine endpoints (if applicable)**: For a function defined over a closed interval, we also need to evaluate the function at the endpoints to determine absolute extrema.

3. **Determine behavior at critical points and endpoints**: The second derivative test \( f''(x) \) or first derivative test can help classify each critical point as a maximum, minimum, or neither.

4. **Assess the behavior as \( x \to \infty \) or \( x \to -\infty \)**: For functions defined on open intervals, we examine limits to infer whether the function has absolute extrema at infinity or if it lacks them.

### Now, considering the choices:
- **Choice A** describes a situation where the function has an absolute maximum at a particular point, but no absolute minimum.
- **Choice B** describes a function with an absolute minimum, but no absolute maximum.
- **Choice C** describes a function that has both an absolute minimum and maximum at specific points.
- **Choice D** indicates no local extreme points at all.

To proceed with solving this, I would need either the function itself or information on its derivative or graphical behavior. Could you provide that, or clarify further? That will help in selecting the correct option.

---

### Possible Follow-up Questions:
1. What is the function \( f(x) \) you're working with?
2. Have you calculated the first and second derivatives of the function?
3. Is the function defined over a closed interval or over \( (-\infty, \infty) \)?
4. Do you know if the function has any critical points?
5. Are there any given limits as \( x \to \infty \) or \( x \to -\infty \)?

### Tip:
When determining extrema, always check the endpoints of the domain (if applicable) along with critical points inside the domain.

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. The function f has an absolute maximum of ___ at x = ___ and no absolute minimum. (Type your answers using exponential notation. Use a comma to separate answers as needed.)

B. The function f has an absolute minimum of ___ at x = ___ and no absolute maximum. (Type your answers using exponential notation. Use a comma to separate answers as needed.)

C. The function f has an absolute minimum of ___ at x = ___ and an absolute maximum of ___ at x = ___. (Type your answers using exponential notation. Use a comma to separate answers as needed.)

D. There are no local extreme points.

Learn how to analyze a function to find its absolute maximum, minimum, and local extreme points using calculus techniques like the first and second derivative tests. This problem includes critical points analysis and is suitable for students in Grades 11-12 and college-level calculus.

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