Math Problem Statement
Solution
We are given the function on the interval . To find the absolute maximum and minimum values, we'll follow these steps:
1. Find the critical points:
We first take the derivative of to find where the slope is zero (critical points).
Set the derivative equal to zero to find critical points:
So, is a critical point.
2. Evaluate the function at the endpoints and critical points:
The function must also be evaluated at the boundaries of the interval, and , and at the critical point .
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At :
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At :
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At :
3. Determine the maximum and minimum values:
Now, we compare the values of at the critical point and endpoints:
The absolute maximum value is , and this occurs at .
The absolute minimum value is , and this occurs at .
Final Answers:
- Absolute maximum value: at .
- Absolute minimum value: at .
Would you like more details on any step or have any questions?
Here are 5 questions related to this topic:
- How does the second derivative help in classifying critical points?
- What is the geometric interpretation of finding maximum and minimum values on a given interval?
- Can a function have more than one absolute maximum or minimum value?
- How do constraints on an interval affect the evaluation of extrema?
- Why is it important to check both critical points and endpoints in a closed interval?
Tip: Always check both the critical points and the endpoints when finding absolute extrema on a closed interval.
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Math Problem Analysis
Mathematical Concepts
Calculus
Optimization
Derivatives
Critical Points
Formulas
f(x) = 4 - 6x^2
f'(x) = -12x
f'(x) = 0 for critical points
Theorems
First Derivative Test
Extreme Value Theorem
Suitable Grade Level
Grades 11-12
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