Math Problem Statement
Solution
To solve this problem, we'll apply the Mean Value Theorem (MVT). According to the MVT, for a function that is continuous on and differentiable on , there exists at least one point such that:
Given Information:
- Function:
- Interval:
Step 1: Calculate and .
Here, and .
Step 2: Compute the average rate of change.
Step 3: Find .
Differentiate :
Step 4: Set and solve for .
Final Answer:
The value of that satisfies the Mean Value Theorem for the given function and interval is .
Would you like a deeper explanation of any of these steps, or do you have any questions?
Related Questions:
- What are the conditions for applying the Mean Value Theorem?
- How do you find the derivative of a quadratic function?
- Can the Mean Value Theorem be applied if the function is not differentiable on the interval?
- What is the geometric interpretation of the Mean Value Theorem?
- How does the Mean Value Theorem relate to Rolle's Theorem?
Tip:
Always verify that the function is continuous on the closed interval and differentiable on the open interval before applying the Mean Value Theorem.
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Math Problem Analysis
Mathematical Concepts
Calculus
Differentiation
Mean Value Theorem
Formulas
f'(x) = d/dx(f(x))
(f(b) - f(a)) / (b - a) = f'(c)
Theorems
Mean Value Theorem
Suitable Grade Level
Grade 12
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