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Extreme Value Theorem

Finding critical points such as relative extrema and absolute extrema.

Mapped to AP College Board # FUN-1, FUN-1.C, FUN-1.C.1, FUN-4, FUN-4.A, FUN-4.A.2, FUN-4.A.3, FUN-4.A.7, FUN-4.A.8

Existence theorems allow us to draw conclusions about a function’s behavior on an interval without precisely locating that behavior. Justify conclusions about functions by applying the Extreme Value Theorem. If a function f is continuous over the interval (a, b), then the Extreme Value Theorem guarantees that f has at least one minimum value and at least one maximum value on (a, b). A function’s derivative can be used to understand some behaviors of the function. Justify conclusions about the behavior of a function based on the behavior of its derivatives. The first derivative of a function can determine the location of relative (local) extrema of the function. Absolute (global) extrema of a function on a closed interval can only occur at critical points or at endpoints. The second derivative of a function may determine whether a critical point is the location of a relative (local) maximum or minimum. When a continuous function has only one critical point on an interval on its domain and the critical point corresponds to a relative (local) extremum of the function on the interval, then that critical point also corresponds to the absolute (global) extremum of the function on the interval.