find the local extrema; any min or max in a graph, absolute extema; the highest or lowest point of a graph, and asymptotes; a line that constantly approaches a curve

To find local extrema, absolute extrema, and asymptotes for a given function or graph, follow these steps:

### 1. **Local Extrema (Local Minimum and Maximum)**
   - **Definition:** Local extrema are points where the function changes direction (from increasing to decreasing or vice versa). These points are known as local maxima (highest point in a neighborhood) and local minima (lowest point in a neighborhood).
   - **How to Find:**
     1. **Take the first derivative** of the function \( f'(x) \).
     2. **Set the first derivative to zero**: Solve \( f'(x) = 0 \) to find the critical points.
     3. **Determine the nature of each critical point** by using the second derivative test or by analyzing the sign of \( f'(x) \) around the critical points.
        - **Second Derivative Test:** If \( f''(x) > 0 \), the point is a local minimum. If \( f''(x) < 0 \), the point is a local maximum.

### 2. **Absolute Extrema (Global Maximum and Minimum)**
   - **Definition:** Absolute extrema are the highest and lowest points on the entire graph of the function.
   - **How to Find:**
     1. Identify all the critical points (as found in the local extrema step).
     2. Evaluate the function \( f(x) \) at each critical point.
     3. Also, evaluate the function at the endpoints of the domain (if the domain is bounded).
     4. The highest value among these points is the **absolute maximum**, and the lowest is the **absolute minimum**.

### 3. **Asymptotes**
   - **Definition:** Asymptotes are lines that the graph of a function approaches but never touches.
   - **Types of Asymptotes:**
     1. **Vertical Asymptotes:** Occur when the function approaches infinity as \( x \) approaches a specific value.
        - **How to Find:** Set the denominator of the function equal to zero and solve for \( x \) (if the function is rational).
     2. **Horizontal Asymptotes:** Occur when the function approaches a constant value as \( x \) approaches infinity or negative infinity.
        - **How to Find:** Analyze the limits \( \lim_{x \to \infty} f(x) \) and \( \lim_{x \to -\infty} f(x) \).
     3. **Oblique Asymptotes:** Occur when the function approaches a line that isn't horizontal or vertical as \( x \) approaches infinity.
        - **How to Find:** If the degree of the numerator is one more than the degree of the denominator in a rational function, perform polynomial division to find the equation of the asymptote.

### Example (Summary of Steps for a Specific Function)
Suppose \( f(x) = \frac{2x^2 - 8}{x - 3} \):

1. **Find Local Extrema:**
   - Differentiate \( f(x) \), find critical points, and use the second derivative test.
   
2. **Find Absolute Extrema:**
   - Evaluate \( f(x) \) at critical points and endpoints if the domain is bounded.
   
3. **Find Asymptotes:**
   - Vertical: Solve \( x - 3 = 0 \).
   - Horizontal: Analyze \( \lim_{x \to \infty} f(x) \).
   - Oblique: Perform polynomial division if applicable.

Would you like a detailed example with a specific function or have any further questions?

### 5 Related Questions
1. How do you find critical points of a function in more complex cases?
2. What is the significance of the second derivative in determining concavity?
3. How do you handle cases where a function has no horizontal asymptotes but does have oblique asymptotes?
4. What are the implications of a function having multiple local extrema but no absolute extrema?
5. How do vertical asymptotes differ from points of discontinuity?

### Tip
Always check for domain restrictions before finding extrema or asymptotes to avoid incorrect conclusions.

Learn how to find local extrema, absolute extrema, and asymptotes in graphs with detailed steps and examples. This guide covers key calculus concepts including the first and second derivative tests, as well as limits. Suitable for advanced high school students and college-level calculus learners.

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