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Questions and Answers
What does the term 'critical number' refer to in calculus?
What does the term 'critical number' refer to in calculus?
If the derivative of a function changes from negative to positive at a critical number, what can be concluded about the function at that point?
If the derivative of a function changes from negative to positive at a critical number, what can be concluded about the function at that point?
What does a horizontal tangent line at a point on a curve indicate?
What does a horizontal tangent line at a point on a curve indicate?
If a function's first derivative does not change sign around a critical number, what does this imply?
If a function's first derivative does not change sign around a critical number, what does this imply?
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Given $f(x) = x^2 - 6x + 5$, what is the x-coordinate of the local minimum?
Given $f(x) = x^2 - 6x + 5$, what is the x-coordinate of the local minimum?
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Given $f(x) = -x^3 + 3x$, which of the following is true concerning local extrema?
Given $f(x) = -x^3 + 3x$, which of the following is true concerning local extrema?
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For a function $f(x)$, its first derivative is $f'(x) = 4x^3 - 12x^2$. Which of the following describes the function’s behavior at a critical number?
For a function $f(x)$, its first derivative is $f'(x) = 4x^3 - 12x^2$. Which of the following describes the function’s behavior at a critical number?
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Which step in the process of finding local extrema involves using the critical numbers to determine where a function is increasing or decreasing?
Which step in the process of finding local extrema involves using the critical numbers to determine where a function is increasing or decreasing?
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Study Notes
Finding Local Extrema
- Local Maximum/Minimum: A point on a curve where the function reaches its highest or lowest value within a specific interval.
- Horizontal Tangent Line: At local extrema points, there's a horizontal tangent line, meaning the derivative of the function is zero at these points.
- Critical Numbers: Values of x where the first derivative of the function is equal to zero or undefined. These values are crucial for identifying potential local extrema points.
- Sign Chart: Used to analyze the behavior of the function's derivative around critical numbers.
- Derivative changes from negative to positive: Local minimum at that critical number.
- Derivative changes from positive to negative: Local maximum at that critical number.
- No sign change: The critical number does not indicate a local extremum.
- Finding Local Extrema: Procedure
- Calculate the first derivative of the function.
- Set the first derivative equal to zero and solve for x to find critical numbers.
- Construct a sign chart using the critical numbers.
- Analyze the change in sign of the first derivative around each critical number to identify local maxima and minima.
- Evaluate the function at the x-values of the local extrema to find the corresponding y-values, providing the coordinates of these extrema.
Example 1: Finding Local Extrema for f(x) = x² - 4x
- First Derivative: f'(x) = 2x - 4
- Critical Number: Setting f'(x) = 0, x = 2.
- Sign Chart:
- For x < 2, f'(x) is negative; the function is decreasing.
- For x > 2, f'(x) is positive; the function is increasing.
- Local minimum at x = 2.
- Local Minimum Value: f(2) = -4; Local minimum at (2, -4).
Example 2: Finding Local Extrema for f(x) = 2x³ + 3x² - 12x
- First Derivative: f'(x) = 6x² + 6x - 12
- Critical Numbers: Factoring f'(x) gives (6)(x + 2)(x - 1). Setting each factor to zero gives x = -2 and x = 1.
- Sign Chart:
- For x > 1, f'(x) is positive.
- For x between -2 and 1, f'(x) is negative.
- For x < -2, f'(x) is positive.
- Local Extrema:
- Local maximum at x = -2.
- Local minimum at x = 1.
- Ordered Pair Representation:
- Local maximum: (-2, 20)
- Local minimum: (1, -7)
Example 3: Finding Local Extrema for f(x) = 3x⁴ - 16x³ + 24x²
- First Derivative: f'(x) = 12x³ - 48x² + 48x
- Critical Numbers: Factoring f'(x) yields 12x(x - 2)² . Setting each factor to zero gives x = 0 and x = 2.
- Sign Chart:
- For x > 2, f'(x) is positive.
- For x between 0 and 2, f'(x) is positive.
- For x < 0, f'(x) is negative.
- Local Extrema:
- Local minimum at x = 0.
- No local extremum at x = 2 (derivative does not change sign).
- Conclusion: The function has a local minimum at x = 0.
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Description
This quiz focuses on the concepts of local maximum and minimum points in calculus. You'll learn how to identify critical numbers, analyze the behavior of derivatives, and use sign charts effectively. Test your understanding of how to find local extrema in various functions.
