Math Problem Statement
Remember this statememnt can possibly be answered as "either true or false"
Solution
To determine the validity of the statement, we can apply the Mean Value Theorem (MVT):
Mean Value Theorem:
If a function is continuous on a closed interval and differentiable on the open interval , then there exists at least one such that:
Given Data:
- The interval is , where is continuous and differentiable.
- Function values:
- The statement asks if there exists such that .
Step 1: Compute the Average Rate of Change
To check if is achievable, calculate the average rate of change of on the entire interval :
Thus, the average rate of change over the interval is 7.
Step 2: Compare to 7
- , which is less than 7.
By the Mean Value Theorem, there exists at least one point in the interval where , the average rate of change. However, the statement specifically asks for , which is not equal to 7.
Step 3: Analyze the Statement
- Since the function is differentiable, can take different values within the interval.
- However, the Mean Value Theorem does not guarantee that because it only ensures the existence of one where equals the average rate of change.
Thus, the statement:
"There exists a value such that "
cannot be guaranteed.
Final Answer:
The statement must be false.
Do you want me to expand on the Mean Value Theorem or provide more examples? Let me know!
Related Questions:
- What is the formal definition of the Mean Value Theorem (MVT)?
- How is the average rate of change of a function calculated?
- Can take multiple values within an interval?
- Under what conditions does the MVT apply?
- How is the Mean Value Theorem used to solve real-world problems?
Tip:
The Mean Value Theorem connects the average rate of change of a function to the instantaneous rate of change at some point, making it a fundamental result in calculus.
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Math Problem Analysis
Mathematical Concepts
Mean Value Theorem
Differentiation
Average Rate of Change
Formulas
Mean Value Theorem: f'(c) = (f(b) - f(a)) / (b - a)
Average Rate of Change: (f(b) - f(a)) / (b - a)
Theorems
Mean Value Theorem
Suitable Grade Level
Grades 11-12 (Calculus)
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