Math Problem Statement
The function ff is defined by f, of, x, equals, x, squared, minus, 4, plus, 3, cosine, left parenthesis, 2, x, right parenthesis, .f(x)=x 2 −4+3cos(2x). Find all values of xx that satisfy the conclusion of the Mean Value Theorem on the interval open square bracket, minus, 2, point, 5, comma, 4, close square bracket, .[−2.5,4]. You may use a calculator and round to the nearest thousandth.
Solution
The Mean Value Theorem (MVT) states that if a function is continuous on a closed interval and differentiable on the open interval , there exists at least one value in such that:
Given Information:
- Function: ,
- Interval: .
Step 1: Verify continuity and differentiability
The function is a combination of polynomial and trigonometric terms, which are continuous and differentiable everywhere. Thus, meets the conditions of the MVT.
Step 2: Calculate the average rate of change
The average rate of change of over is:
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Compute : Using a calculator:
-
Compute : Using a calculator:
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Average rate of change:
Step 3: Solve
The derivative of is: We solve:
Rearrange:
Step 4: Use numerical methods
Using a calculator or numerical solver, solve this equation in the interval .
- Graphical/numerical solution: By plotting , we find the roots.
After computation, the approximate solutions (rounded to the nearest thousandth) are:
Final Answer:
The values of that satisfy the Mean Value Theorem on are:
Would you like more details about any of these steps?
Related Questions:
- What does the Mean Value Theorem imply geometrically?
- Can we always apply the MVT to a given function?
- How do trigonometric derivatives behave for functions like ?
- Why do numerical methods sometimes produce approximate solutions?
- How does the cosine function affect the behavior of in this problem?
Tip:
When solving , graphing or numerical solvers are often the most efficient approach.
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Math Problem Analysis
Mathematical Concepts
Calculus
Mean Value Theorem
Differentiation
Trigonometric Functions
Formulas
f'(x) = 2x - 6sin(2x)
f'(c) = (f(b) - f(a)) / (b - a)
Theorems
Mean Value Theorem
Suitable Grade Level
Grades 11-12