Math Problem Statement
Verify Lagrange mean value for f(x) = x ^ 3 + x ^ 2 in [1,2]
Solution
To verify the Lagrange Mean Value Theorem (LMVT) for the function on the interval , let's proceed step by step.
Step 1: Verify Conditions of LMVT
LMVT states that if a function is:
- Continuous on a closed interval ,
- Differentiable on an open interval ,
then there exists at least one point such that:
The function is a polynomial, which is continuous and differentiable everywhere, so it satisfies both conditions on .
Step 2: Calculate and
Let's find and to compute the slope of the secant line between these points:
Step 3: Calculate the Secant Slope
The secant slope between and is:
Step 4: Find
Now, we calculate the derivative :
Step 5: Set and Solve for
We now need to find such that : Rearrange to form a standard quadratic equation:
Solve this quadratic equation using the quadratic formula:
Calculating these values: c \approx 1.62 \quad \text{(since 1 < c < 2)}
Conclusion
There exists a point in the interval where , thus satisfying the Lagrange Mean Value Theorem.
Would you like more details or have any questions?
Here are some related questions:
- How does the Mean Value Theorem differ from Rolle's Theorem?
- Can LMVT be applied to trigonometric functions over specified intervals?
- What are the implications of LMVT in real-world scenarios?
- How does LMVT apply to non-polynomial functions?
- How can we approximate values using LMVT in calculus problems?
Tip: Always check the continuity and differentiability of the function over the interval to verify conditions for applying LMVT.
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Math Problem Analysis
Mathematical Concepts
Calculus
Mean Value Theorem
Derivatives
Polynomials
Formulas
Lagrange Mean Value Theorem formula: f'(c) = (f(b) - f(a)) / (b - a)
Derivative of f(x) = x^3 + x^2: f'(x) = 3x^2 + 2x
Theorems
Lagrange Mean Value Theorem
Suitable Grade Level
Grades 11-12
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