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- https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_(Boman_and_Rogers)/07%3A_Intermediate_and_Extreme_Values/7.04%3A_The_Supremum_and_the_Extreme_Value_TheoremA continuous function on a closed, bounded interval must be bounded. Boundedness, in and of itself, does not ensure the existence of a maximum or minimum. We must also have a closed, bounded interval.
- https://math.libretexts.org/Courses/Reedley_College/Calculus_I_(Casteel)/04%3A_Applications_of_Derivatives/4.02%3A_Maxima_and_MinimaFinding the maximum and minimum values of a function has practical significance because we can use this method to solve optimization problems, such as maximizing profit, minimizing the amount of mate...Finding the maximum and minimum values of a function has practical significance because we can use this method to solve optimization problems, such as maximizing profit, minimizing the amount of material used in manufacturing an aluminum can, or finding the maximum height a rocket can reach. In this section, we look at how to use derivatives to find the largest and smallest values for a function.
- https://math.libretexts.org/Courses/Chabot_College/MTH_1%3A_Calculus_I/04%3A_Applications_of_Derivatives/4.01%3A_Maxima_and_MinimaFinding the maximum and minimum values of a function has practical significance because we can use this method to solve optimization problems, such as maximizing profit, minimizing the amount of mate...Finding the maximum and minimum values of a function has practical significance because we can use this method to solve optimization problems, such as maximizing profit, minimizing the amount of material used in manufacturing an aluminum can, or finding the maximum height a rocket can reach. In this section, we look at how to use derivatives to find the largest and smallest values for a function.
- https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/Interactive_Calculus_Q1/04%3A_Applications_of_Derivatives/4.04%3A_Maxima_and_MinimaFinding the maximum and minimum values of a function has practical significance because we can use this method to solve optimization problems, such as maximizing profit, minimizing the amount of mate...Finding the maximum and minimum values of a function has practical significance because we can use this method to solve optimization problems, such as maximizing profit, minimizing the amount of material used in manufacturing an aluminum can, or finding the maximum height a rocket can reach. In this section, we look at how to use derivatives to find the largest and smallest values for a function.
- https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_(Boman_and_Rogers)/07%3A_Intermediate_and_Extreme_Values/7.E%3A_Intermediate_and_Extreme_Values_(Exercises)Mimic the definitions of an upper bound of a set and the least upper bound (supremum) of a set to give definitions for a lower bound of a set and the greatest lower bound (infimum) of a set. Find the ...Mimic the definitions of an upper bound of a set and the least upper bound (supremum) of a set to give definitions for a lower bound of a set and the greatest lower bound (infimum) of a set. Find the least upper bound (supremum) and greatest lower bound (infimum) of the following sets of real numbers, if they exist. (If one does not exist then say so.) Prove that b is an upper bound of S if and only if −b is a lower bound of T.
- https://math.libretexts.org/Courses/Coastline_College/Math_C180%3A_Calculus_I_(Tran)/04%3A_Applications_of_Derivatives/4.03%3A_Maxima_and_MinimaFinding the maximum and minimum values of a function has practical significance because we can use this method to solve optimization problems, such as maximizing profit, minimizing the amount of mate...Finding the maximum and minimum values of a function has practical significance because we can use this method to solve optimization problems, such as maximizing profit, minimizing the amount of material used in manufacturing an aluminum can, or finding the maximum height a rocket can reach. In this section, we look at how to use derivatives to find the largest and smallest values for a function.
- https://math.libretexts.org/Courses/Coastline_College/Math_C180%3A_Calculus_I_(Everett)/04%3A_Applications_of_Derivatives/4.03%3A_Maxima_and_MinimaFinding the maximum and minimum values of a function has practical significance because we can use this method to solve optimization problems, such as maximizing profit, minimizing the amount of mate...Finding the maximum and minimum values of a function has practical significance because we can use this method to solve optimization problems, such as maximizing profit, minimizing the amount of material used in manufacturing an aluminum can, or finding the maximum height a rocket can reach. In this section, we look at how to use derivatives to find the largest and smallest values for a function.
- https://math.libretexts.org/Courses/Prince_Georges_Community_College/MAT_2410%3A_Calculus_1_(Beck)/04%3A_Applications_of_Derivatives/4.04%3A_Maxima_and_MinimaFinding the maximum and minimum values of a function has practical significance because we can use this method to solve optimization problems, such as maximizing profit, minimizing the amount of mate...Finding the maximum and minimum values of a function has practical significance because we can use this method to solve optimization problems, such as maximizing profit, minimizing the amount of material used in manufacturing an aluminum can, or finding the maximum height a rocket can reach. In this section, we look at how to use derivatives to find the largest and smallest values for a function.
- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_210_Calculus_I_(Professor_Dean)/Chapter_4%3A_Applications_of_Derivatives/4.2%3A_Maxima_and_MinimaFrom the graph of f in Figure, we see that f has an absolute maximum at x=1 and an absolute minimum at x=−1. Hence, f has a local maximum at x=1 and a local minimum at x=−1. (N...From the graph of f in Figure, we see that f has an absolute maximum at x=1 and an absolute minimum at x=−1. Hence, f has a local maximum at x=1 and a local minimum at x=−1. (Note that if f has an absolute extremum over an interval I at a point c that is not an endpoint of I, then f has a local extremum at c.)
- https://math.libretexts.org/Courses/City_College_of_San_Francisco/CCSF_Calculus/04%3A_Appropriate_Applications/4.01%3A_Graphing_Using_Calculus_-_Maxima_and_MinimaThis section covers how to use calculus to graph functions by finding critical points, maxima, and minima. It explains how the first and second derivative tests help identify increasing or decreasing ...This section covers how to use calculus to graph functions by finding critical points, maxima, and minima. It explains how the first and second derivative tests help identify increasing or decreasing intervals and concavity, which are key to determining the shape of the graph. The section also demonstrates how to apply these concepts to real-world problems, optimizing functions to find their maximum or minimum values.
- https://math.libretexts.org/Under_Construction/Purgatory/Book%3A_Active_Calculus_(Boelkins_et_al.)/03%3A_Using_Derivatives/3.03%3A_Global_OptimizationTo find relative extreme values of a function, we normally use a first derivative sign chart and classify all of the function’s critical numbers. If instead we are interested in absolute extreme value...To find relative extreme values of a function, we normally use a first derivative sign chart and classify all of the function’s critical numbers. If instead we are interested in absolute extreme values, we first decide whether we are considering the entire domain of the function or a particular interval. If we are working to find absolute extremes on a restricted interval, then we first identify all critical numbers of the function that lie in the interval
