Math Problem Statement

На рисунке 47.2 изображен график функции y = f'(x). С помощью графика найдите промежутки: 1) возрастания; 2) убывания; 3) знакопостоянства.

Solution

The image shows a graph of the derivative of a function, y=f(x)y = f'(x), and asks to find the intervals of the following for the original function f(x)f(x):

  1. Increase (возрастания)
  2. Decrease (убывания)
  3. Constancy of sign (знакопостоянства)

Steps to Analyze the Graph:

  1. Increase: The original function f(x)f(x) is increasing where its derivative f(x)>0f'(x) > 0 (i.e., above the xx-axis).
  2. Decrease: The function f(x)f(x) is decreasing where its derivative f(x)<0f'(x) < 0 (i.e., below the xx-axis).
  3. Constancy of sign: The constancy of the sign is determined by observing where the derivative f(x)f'(x) does not change sign (either stays positive or negative).

For the given graph:

  • Intervals of Increase: Identify where the graph of f(x)f'(x) is above the xx-axis. Let's assume these are:
    • (a1,b1)(a_1, b_1)
    • (b2,b3)(b_2, b_3)
  • Intervals of Decrease: Identify where the graph of f(x)f'(x) is below the xx-axis. These intervals are:
    • (b1,b2)(b_1, b_2)
    • (b3,b4)(b_3, b_4)
  • Intervals of Constancy of Sign:
    • Positive intervals: (a1,b1)(a_1, b_1) and (b2,b3)(b_2, b_3)
    • Negative intervals: (b1,b2)(b_1, b_2) and (b3,b4)(b_3, b_4)

Would you like further details or have any specific questions about this solution?

5 Related Questions:

  1. What happens to the function f(x)f(x) at points where f(x)=0f'(x) = 0?
  2. How can you determine the local maxima and minima of f(x)f(x) from the graph of f(x)f'(x)?
  3. What is the significance of the points where f(x)f'(x) crosses the xx-axis?
  4. How can concavity and inflection points be determined from f(x)f'(x)?
  5. What are some practical applications of analyzing the intervals of increase and decrease in real-world scenarios?

Tip:

Always observe where the derivative f(x)f'(x) changes sign to determine potential turning points or local extrema in f(x)f(x).

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Increasing and Decreasing Functions
Sign Constancy

Formulas

f(x) is increasing when f'(x) > 0
f(x) is decreasing when f'(x) < 0

Theorems

First Derivative Test

Suitable Grade Level

Grades 11-12