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Let's analyze the given function:

### Problem:
You are asked to determine the relative maxima or minima of the function \( f(x) = -x^2 + 13x - 36 \), and the intervals on which the function is increasing or decreasing.

### Solution:

1. **Step 1: Find the first derivative**  
   The first derivative of the function helps us find critical points, which are candidates for relative maxima and minima.
   \[
   f'(x) = \frac{d}{dx} \left( -x^2 + 13x - 36 \right) = -2x + 13
   \]

2. **Step 2: Set the derivative equal to zero to find critical points**  
   Set \( f'(x) = 0 \) to find where the slope is zero, indicating possible maxima or minima.
   \[
   -2x + 13 = 0
   \]
   Solving for \( x \):
   \[
   x = \frac{13}{2} = 6.5
   \]

3. **Step 3: Determine whether it's a maximum or minimum**  
   Use the second derivative or test values around \( x = 6.5 \) to determine if this critical point is a maximum or minimum. The second derivative is:
   \[
   f''(x) = \frac{d}{dx} \left( -2x + 13 \right) = -2
   \]
   Since the second derivative is negative (\( f''(x) = -2 < 0 \)), the function is concave down at \( x = 6.5 \), indicating a **relative maximum** at this point.

4. **Step 4: Find the value of the function at \( x = 6.5 \)**  
   Substitute \( x = 6.5 \) into the original function to find the relative maximum point:
   \[
   f(6.5) = -(6.5)^2 + 13(6.5) - 36 = -42.25 + 84.5 - 36 = 6.25
   \]
   So, the relative maximum occurs at \( (6.5, 6.25) \).

5. **Step 5: Determine intervals of increase and decrease**  
   - For \( f'(x) = -2x + 13 \), check the sign of \( f'(x) \):
     - When \( x < 6.5 \), \( f'(x) > 0 \), so the function is increasing.
     - When \( x > 6.5 \), \( f'(x) < 0 \), so the function is decreasing.

### Conclusion:
- The function has a **relative maximum** at \( x = 6.5 \).
- The function is **increasing** on the interval \( (-\infty, 6.5) \).
- The function is **decreasing** on the interval \( (6.5, \infty) \).

Let me know if you would like further details or have any questions!

Here are 5 related questions you could explore:
1. How does the second derivative test confirm maxima or minima?
2. What happens if both the first and second derivatives are zero at a critical point?
3. Can a function have both relative maxima and minima?
4. How do concavity and inflection points relate to relative extrema?
5. What are some real-world applications of finding maxima and minima in functions?

**Tip**: Always double-check your derivative and second derivative calculations when solving for extrema and concavity!

Determine any relative maxima or minima of the function and the intervals on which the function is increasing or decreasing. f(x) = -x^2 + 13x - 36

Learn how to determine the relative maxima and minima of the function f(x) = -x^2 + 13x - 36, along with identifying the increasing and decreasing intervals using calculus methods. This problem involves taking the first and second derivatives and using them to analyze the function behavior. Suitable for students in Grades 10-12.

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