Math Problem Statement
Solution
The problem asks to find the value of where the local minimum of the function occurs.
Steps to solve:
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Find the derivative of : The derivative helps identify critical points where the slope of the function is zero (possible local minima or maxima).
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Set the derivative equal to zero: To find critical points: Simplify:
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Determine the nature of the critical points: Use the second derivative to check whether each critical point is a local minimum or maximum.
The second derivative is:
- For :
- For :
Thus, the local minimum occurs at .
The correct response is Response 2: .
Would you like further details on any steps or concepts involved in this solution?
Here are five related questions to deepen your understanding:
- How do second derivatives help in identifying local minima and maxima?
- What are the necessary conditions for a critical point to be a saddle point?
- Can a cubic function have more than one local minimum?
- How does concavity relate to the second derivative of a function?
- How would the solution change if the function had higher-order terms?
Tip: Always check critical points using the second derivative test to confirm whether they are minima, maxima, or saddle points.
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Math Problem Analysis
Mathematical Concepts
Calculus
Critical Points
Derivatives
Second Derivative Test
Formulas
f'(x) = 3x^2 - 3 (First Derivative)
f''(x) = 6x (Second Derivative)
Local minimum condition: f''(x) > 0
Theorems
First Derivative Test
Second Derivative Test
Suitable Grade Level
Grades 10-12
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