Math Problem Statement
Solution
To determine the absolute maximum value of on the closed interval , follow these steps:
Step 1: Find the derivative of
We compute to determine the critical points:
Step 2: Solve to find critical points
Set : Divide through by 3: Factorize:
Thus, is the only critical point in the interval .
Step 3: Evaluate at endpoints and critical points
We now compute at , , and .
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At :
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At (critical point):
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At :
Step 4: Compare the values
From the calculations:
The absolute maximum value occurs at , and the maximum value is:
Final Answer:
The absolute maximum value of on the interval is 18, and it occurs at .
Would you like further clarifications or help with a similar problem? Here are 5 related questions:
- How do you find absolute minimum values on a closed interval?
- What is the difference between critical points and extrema?
- How do you handle endpoints when finding absolute extrema?
- How do derivatives help in identifying critical points?
- Can you apply the same process for trigonometric or exponential functions?
Tip: Always check endpoints when finding absolute maxima or minima on a closed interval!
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Math Problem Analysis
Mathematical Concepts
Calculus
Critical Points
Absolute Extrema
Derivative Test
Formulas
f'(x) = d/dx [f(x)]
f'(x) = 0 to find critical points
Theorems
Extreme Value Theorem
Suitable Grade Level
Grades 11-12 (Advanced Calculus)