AP Calculus AB Flashcards: Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
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What two conditions must be met for the Extreme Value Theorem to guarantee the existence of a global maximum and minimum?
The function must be continuous, and the interval must be closed (e.g., in the form $[a, b]$).
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What two conditions must be met for the Extreme Value Theorem to guarantee the existence of a global maximum and minimum?
The function must be continuous, and the interval must be closed (e.g., in the form $[a, b]$).
How are the critical points of a function $f$ identified?
Critical points are identified by finding all points in the function's domain where the first derivative, $f'(x)$, is equal to zero or where $f'(x)$ does not exist.
Where must all local maximums and minimums of a function be located?
All local (relative) extrema must be located at the critical points of the function.
What is the Extreme Value Theorem (EVT)?
If a function $f$ is continuous over a closed interval $[a, b]$, then the EVT guarantees that $f$ has at least one minimum value and at least one maximum value on $[a, b]$.
How would you use the Extreme Value Theorem to justify a conclusion about a function's absolute extrema?
You would state that because the function is continuous on a closed interval, the Extreme Value Theorem guarantees the existence of at least one absolute minimum and one absolute maximum.
A function $g(x)$ is continuous on the interval [0, 10]. What conclusion can be drawn by applying the Extreme Value Theorem?
The Extreme Value Theorem guarantees that $g(x)$ has at least one absolute minimum value and at least one absolute maximum value on the interval [0, 10].
Why can the Extreme Value Theorem not be applied to the function $f(x) = \\tan(x)$ on the interval $[0, \\pi]$?
The EVT cannot be applied because the function is not continuous on the closed interval $[0, \\pi]$, as it has a vertical asymptote at $x = \\pi/2$.
What is the relationship between local (relative) extrema and critical points?
All local (relative) extrema of a function must occur at its critical points.
If a point is a critical point, is it guaranteed to be a local extremum?
No, not all critical points are local extrema. A function can have a critical point that is neither a local maximum nor a local minimum.
What is a critical point of a function?
A critical point is a point on a function where the first derivative either equals zero or fails to exist.