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I see that you've uploaded an image with a graph and some questions written in Russian. I'll analyze the graph and translate the questions to solve them step by step. Let me process this for you.The text and graph describe a mathematical problem based on the derivative \( y = f'(x) \), where you're tasked to analyze the behavior of the function \( y = f(x) \) using its derivative graph over the interval \( (-2, 9) \). Here's the breakdown of the questions:

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### **Translated Questions:**
(a) Find the points of extremum and their type (minimum or maximum).

(b) Identify intervals where the function is increasing and decreasing.

(c) Determine the point on the interval \([-1, 8]\) where the function \( y = f(x) \) reaches its maximum value.

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### **Step-by-Step Explanation and Solution:**

1. **Points of Extremum (a):**
   - Extremum points of \( f(x) \) occur where \( f'(x) = 0 \) (the graph intersects the x-axis) and changes sign.
   - From the graph, \( f'(x) = 0 \) at:
     \[
     x = -1, x = 2, x = 6
     \]
   - To determine the type of extremum:
     - At \( x = -1 \), \( f'(x) \) changes from positive to negative, so it's a **local maximum**.
     - At \( x = 2 \), \( f'(x) \) changes from negative to positive, so it's a **local minimum**.
     - At \( x = 6 \), \( f'(x) \) changes from positive to negative, so it's a **local maximum**.

2. **Intervals of Increase and Decrease (b):**
   - The function \( f(x) \) is increasing where \( f'(x) > 0 \) (above the x-axis).
   - It is decreasing where \( f'(x) < 0 \) (below the x-axis).
   - From the graph:
     - Increasing on \( (-2, -1) \cup (2, 6) \)
     - Decreasing on \( (-1, 2) \cup (6, 9) \)

3. **Maximum Value on \([-1, 8]\) (c):**
   - To find the maximum, consider critical points and endpoints:
     - Critical points from \( f'(x) = 0 \): \( x = -1, x = 2, x = 6 \)
     - Evaluate \( f(x) \) at these points and endpoints \( x = -1, x = 8 \) (direct evaluation not possible here but based on trends).
   - The maximum occurs at \( x = -1 \) or \( x = 6 \), depending on the relative height.

Would you like further clarifications on any part? Below are five related questions to explore further.  

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### Related Questions:
1. How can you determine concavity and points of inflection from \( f'(x) \)?
2. What role does the second derivative \( f''(x) \) play in identifying extrema?
3. How do the endpoints of the interval affect global extrema determination?
4. Can a function have multiple global maxima on a closed interval? Why or why not?
5. How does the graph of \( f'(x) \) relate to the slope of the tangent line to \( f(x) \)?

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### Tip:
Always analyze the sign changes of \( f'(x) \) to confirm extrema and trends of \( f(x) \), and consider endpoints when finding absolute maxima or minima on a closed interval.

Using the graph of the derivative y = f'(x) of the function y = f(x) over the interval (-2,9), determine: (a) the points of extrema and their types, (b) the intervals where the function is increasing and decreasing, and (c) the point on [-1,8] where y = f(x) reaches its maximum.

Learn how to determine extrema, increasing and decreasing intervals, and the maximum value on a given range using the derivative graph f'(x). This step-by-step guide covers key calculus concepts including critical points and the First Derivative Test.

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