Flashcards on Local extrema and inflection points
Local Extrema and Inflection Points: A Complete Guide
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Calculus Function Analysis
27 cards
Card 1
Question: What does the first derivative of a function, f'(x), represent geometrically?
Answer: The slope of the tangent line to the function's graph at that point.
Card 2
Question: What does it mean if f′(x) > 0 on an interval?
Answer: The function f(x) is increasing on that interval.
Card 3
Question: What does it mean if f′(x) < 0 on an interval?
Answer: The function f(x) is decreasing on that interval.
Card 4
Question: What is the behavior of f(x) if f′(x) = 0 on an interval?
Answer: The function f(x) is constant on that interval.
Card 5
Question: Define a local maximum for a function f(x) at a point x = c.
Answer: f(c) is greater than or equal to all nearby function values.
Card 6
Question: Define a local minimum for a function f(x) at a point x = c.
Answer: f(c) is less than or equal to all nearby function values.
Card 7
Question: What are local maximums and local minimums collectively called?
Answer: Local extrema.
Card 8
Question: What are stationary points (critical points)?
Answer: Points where the first derivative f′(x) = 0.
Card 9
Question: Are all stationary points local extrema?
Answer: No, some stationary points can be inflection points.
Card 10
Question: According to the First Derivative Test, what does it mean if f′(x) changes from positive to negative at x = c?
Answer: f(c) is a local maximum.