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Questions and Answers
What is a critical number of a function?
What is a critical number of a function?
- A point where the function has a local maximum
- A point where the function is increasing
- A point where the derivative is equal to zero or is undefined (correct)
- A point where the function is decreasing
Which critical number corresponds to a local minimum for the function f(x) = (3x + 1)^{2/3}?
Which critical number corresponds to a local minimum for the function f(x) = (3x + 1)^{2/3}?
- 0
- 2
- -1/3 (correct)
- -1
In the context of local extrema, what does a horizontal tangent line indicate?
In the context of local extrema, what does a horizontal tangent line indicate?
- A local minimum
- An increasing function
- A critical number (correct)
- A local maximum
For the function f(x) = x^{1/3}, where does it have a local extremum?
For the function f(x) = x^{1/3}, where does it have a local extremum?
Identify the local extremum for the function f(x) = 9 - x^2.
Identify the local extremum for the function f(x) = 9 - x^2.
What behavior is observed at local extrema for the function f(x) = x^2 + 5x - 1?
What behavior is observed at local extrema for the function f(x) = x^2 + 5x - 1?
What is true about critical numbers of the function f(x) = -x^2 + 4x + 2?
What is true about critical numbers of the function f(x) = -x^2 + 4x + 2?
In the function f(x) = x^4 - 2x^2 + 1, which points correspond to local extrema?
In the function f(x) = x^4 - 2x^2 + 1, which points correspond to local extrema?
Where do absolute extrema of a continuous function on the closed interval [a, b] occur?
Where do absolute extrema of a continuous function on the closed interval [a, b] occur?
What characterizes a function that is increasing on an interval I?
What characterizes a function that is increasing on an interval I?
What does it indicate if a critical number c causes the function to change from increasing to decreasing?
What does it indicate if a critical number c causes the function to change from increasing to decreasing?
What happens if the derivative f'(x) has the same sign on both sides of a critical number?
What happens if the derivative f'(x) has the same sign on both sides of a critical number?
What is the local extremum of the function f(x) = |x|?
What is the local extremum of the function f(x) = |x|?
What is a necessary condition for a function to have a local minimum at a critical number?
What is a necessary condition for a function to have a local minimum at a critical number?
What does the Extreme Value Theorem state about a continuous function on a closed, bounded interval?
What does the Extreme Value Theorem state about a continuous function on a closed, bounded interval?
For f(x) = 1/x on the interval [−3, 0) ∪ (0, 3], what is true about its absolute extrema?
For f(x) = 1/x on the interval [−3, 0) ∪ (0, 3], what is true about its absolute extrema?
In the function f(x) = 2x^3 + 9x^2 - 24x - 10, if f' changes from negative to positive at a point, that point is identified as:
In the function f(x) = 2x^3 + 9x^2 - 24x - 10, if f' changes from negative to positive at a point, that point is identified as:
Given the function f(x) = x^4 - 4x^3, which of the following statements is true concerning local extrema?
Given the function f(x) = x^4 - 4x^3, which of the following statements is true concerning local extrema?
What is a critical number of a function f?
What is a critical number of a function f?
What is the outcome if a function is decreasing on an interval?
What is the outcome if a function is decreasing on an interval?
What are the critical numbers of the function f(x) = 2x³ - 3x² - 12x + 5 on the interval [−2, 4]?
What are the critical numbers of the function f(x) = 2x³ - 3x² - 12x + 5 on the interval [−2, 4]?
Which interval does f(x) = x² - 9 attain absolute extrema?
Which interval does f(x) = x² - 9 attain absolute extrema?
At the point x = 0 for f(x) = |x|, what is notable about the derivative?
At the point x = 0 for f(x) = |x|, what is notable about the derivative?
What is the absolute maximum value of f(x) = 1/x on the interval [1, 3]?
What is the absolute maximum value of f(x) = 1/x on the interval [1, 3]?
Flashcards
Critical Number
Critical Number
A value in the function's domain where the derivative is either zero or undefined.
Local Extremum
Local Extremum
A point where a function has a local maximum or minimum value. In relation to a particular neighborhood.
Critical number f'(c)=0
Critical number f'(c)=0
A critical number where the derivative of the function is zero
Critical number f'(c) undefined
Critical number f'(c) undefined
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Local Maximum
Local Maximum
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Local Minimum
Local Minimum
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Find critical numbers
Find critical numbers
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Second Derivative Test
Second Derivative Test
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Local Extremum
Local Extremum
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Absolute Maximum
Absolute Maximum
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Absolute Minimum
Absolute Minimum
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Critical Number
Critical Number
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Extreme Value Theorem
Extreme Value Theorem
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Closed Interval
Closed Interval
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Continuous Function
Continuous Function
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Finding Absolute Extrema
Finding Absolute Extrema
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Local Extremum
Local Extremum
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Critical Number
Critical Number
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Increasing Function
Increasing Function
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Decreasing Function
Decreasing Function
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Local Maximum
Local Maximum
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Local Minimum
Local Minimum
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Absolute Extremum
Absolute Extremum
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Point of Inflection
Point of Inflection
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Study Notes
Applications of Differentiation
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Critical Numbers: A number c in the domain of a function f is called a critical number of f if f'(c) = 0 or f'(c) is undefined.
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Local Extrema: Local extrema occur at critical numbers, where a function changes from increasing to decreasing or vice versa.
- A function can have an extremum at a point where the tangent line is horizontal (f'(x) = 0).
- A function can have an extremum at a corner or a point where the tangent line is vertical (f'(x) is undefined).
- Local extrema are relative maximum or minimum values within a specific interval.
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Absolute Extrema: For a function f defined on a set S of real numbers and a number c ∈ S:
- f(c) is the absolute maximum if f(c) ≥ f(x) for all x ∈ S.
- f(c) is the absolute minimum if f(c) ≤ f(x) for all x ∈ S.
- Absolute extrema can occur at critical numbers or endpoints of an interval.
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Increasing and Decreasing Functions:
- A function f is increasing on an interval I if, for any x₁ and x₂ in I with x₁ < x₂, f(x₁) < f(x₂).
- A function f is decreasing on an interval I if, for any x₁ and x₂ in I with x₁ < x₂, f(x₁) > f(x₂).
- Increasing and decreasing behavior of a function can be determined by examining the sign of the first derivative.
- Critical numbers are points where the first derivative is either zero or undefined and where a function changes from increasing to decreasing or vice versa.
Concavity and Second Derivative Test
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Concavity: The concavity of a function describes how the function bends.
- A function is concave upward on an interval if its graph lies above its tangent lines within that interval.
- A function is concave downward on an interval if its graph lies below its tangent lines within that interval.
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Inflection Points: An inflection point is a point on the graph of a function where the concavity changes.
- At an inflection point, the sign of the second derivative changes.
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Second Derivative Test: This test uses the second derivative to determine the nature of critical points.
- If f''(c) > 0 where f'(c) = 0, then f has a local minimum at x = c.
- If f''(c) < 0 where f'(c) = 0, then f has a local maximum at x = c.
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