Math Problem Statement
Solution
The image shows a graph of the derivative of a function, , and asks to find the intervals of the following for the original function :
- Increase (возрастания)
- Decrease (убывания)
- Constancy of sign (знакопостоянства)
Steps to Analyze the Graph:
- Increase: The original function is increasing where its derivative (i.e., above the -axis).
- Decrease: The function is decreasing where its derivative (i.e., below the -axis).
- Constancy of sign: The constancy of the sign is determined by observing where the derivative does not change sign (either stays positive or negative).
For the given graph:
- Intervals of Increase: Identify where the graph of is above the -axis. Let's assume these are:
- Intervals of Decrease: Identify where the graph of is below the -axis. These intervals are:
- Intervals of Constancy of Sign:
- Positive intervals: and
- Negative intervals: and
Would you like further details or have any specific questions about this solution?
5 Related Questions:
- What happens to the function at points where ?
- How can you determine the local maxima and minima of from the graph of ?
- What is the significance of the points where crosses the -axis?
- How can concavity and inflection points be determined from ?
- What are some practical applications of analyzing the intervals of increase and decrease in real-world scenarios?
Tip:
Always observe where the derivative changes sign to determine potential turning points or local extrema in .
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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
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