Math Problem Statement
To find the absolute minimum value of the function f(x)=xlnx, we need to follow these steps:
Solution
To find the absolute minimum value of the function , follow these steps:
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Domain: Identify the domain of the function. The function is defined for , so the domain of is .
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Critical Points: Find the critical points by taking the derivative and setting it equal to zero. Set the derivative equal to zero:
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Second Derivative Test: Use the second derivative to determine whether this critical point is a minimum or maximum. At : Since the second derivative is positive, is a local minimum.
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Value at Critical Point: Evaluate the function at the critical point to find the minimum value.
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Boundary Behavior: Check the behavior of as and .
- As : because and approaches 0.
- As : because grows slower than , making grow to .
Given the analysis, the absolute minimum value of the function is , which occurs at .
Would you like more details on any step or have any questions?
Here are some related questions to consider:
- What is the domain of the natural logarithm function?
- How do you find critical points of a function?
- What is the significance of the second derivative test?
- What happens to as approaches 0 from the right?
- Why does the function approach as increases?
- What are some common techniques to determine the behavior of a function at infinity?
- How does the behavior of differ from polynomial and exponential functions?
- What are the differences between local and absolute extrema of a function?
Tip: Always check the domain and boundary behavior of the function to understand its overall behavior.
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Critical Points
Second Derivative Test
Limit Analysis
Formulas
Derivative formula
Second derivative test formula
Theorems
Second derivative test
Suitable Grade Level
Undergraduate
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