The Extreme Value Theorem ensures a continuous function on a closed interval has both a global maximum and a global minimum value within that interval.
True
Local extrema are the relative maximum and minimum values within a smaller region of the function's domain.
To identify global extrema, you must scan the graph for the highest and lowest points.
To find local extrema, first locate the critical points.
The Extreme Value Theorem states that if a function f(x) is continuous on a closed interval [a, b], then f(x) must attain both a global maximum and a global minimum value somewhere within that interval
Global extrema must occur at critical points or endpoints of the interval.
True
The Extreme Value Theorem applies when a function is continuous on a closed interval.
Match the type of extrema with its characteristic:
Global Extrema ↔️ Absolute max/min over entire domain
Local Extrema ↔️ Relative max/min within a region
Steps to identify global extrema on a graph
1️⃣ Scan the graph for highest and lowest points
2️⃣ Global maximum is the highest point
3️⃣ Global minimum is the lowest point
Critical points, where f'(x) = 0 or is undefined, are potential locations for local extrema.
True
What is the key difference between global and local extrema?
Absolute vs. relative values
Global extrema are defined as the absolute maximum or minimum values over the entire domain
The Extreme Value Theorem applies when a function f(x) is continuous on a closed interval [a, b].
What are global extrema of a function?
Absolute maximum and minimum
Local extrema are always global extrema within their region.
False
Where does the global maximum occur on the graph of f(x) = x^3 - 3x^2 + 2?
Highest point
Steps to identify local extrema on a graph
1️⃣ Find the critical points where f'(x) = 0 or is undefined
2️⃣ Evaluate f(x) at the critical points
3️⃣ Compare neighboring values to determine local maxima or minima