Math Problem Statement
Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval?
f(x) = x3 − 3x +3, [−2, 2] If it satisfies the hypotheses, find all numbers c that satisfy the conclusion of the Mean Value Theorem.
Solution
To determine if the function satisfies the hypotheses of the Mean Value Theorem (MVT) on the interval , we proceed as follows:
Step 1: Verify the hypotheses of the MVT
The Mean Value Theorem requires that:
- is continuous on ,
- is differentiable on .
1. Continuity:
The function is a polynomial. Polynomials are continuous everywhere. Therefore, is continuous on .
2. Differentiability:
The function is a polynomial. Polynomials are differentiable everywhere. Therefore, is differentiable on .
Since both conditions are satisfied, the function satisfies the hypotheses of the Mean Value Theorem on .
Step 2: Apply the Mean Value Theorem
The MVT guarantees that there exists at least one such that: Here, and . Compute the values of and : The slope of the secant line is:
Thus, we need to find such that:
Step 3: Find such that
First, compute : Set to solve for : Simplify: Take the square root:
Step 4: Verify is in the interval
The solutions are: Both values are in the interval .
Final Answer
The function satisfies the hypotheses of the Mean Value Theorem on . The values of that satisfy the conclusion of the theorem are:
Would you like more details or clarification on any step?
Here are 5 related questions for practice:
- How do you prove a function is continuous and differentiable?
- Can the Mean Value Theorem apply to functions that are not differentiable everywhere?
- What does the Mean Value Theorem geometrically represent?
- How would you apply the Mean Value Theorem to a trigonometric function?
- What changes in the process if the interval is not symmetric about the origin?
Tip: For polynomials, continuity and differentiability are always guaranteed! Focus on finding the slope and derivative conditions for MVT.
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Math Problem Analysis
Mathematical Concepts
Calculus
Mean Value Theorem
Polynomials
Continuity
Differentiability
Formulas
Mean Value Theorem: f'(c) = (f(b) - f(a)) / (b - a)
Derivative of a polynomial: d/dx(x^n) = n*x^(n-1)
Theorems
Mean Value Theorem
Suitable Grade Level
Undergraduate Calculus (Grade 11-12 or College)
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