Glossary

13.3 Closed interval method

3 min read•july 22, 2024

The is a game-changer for finding the highest and lowest points of continuous functions. It guarantees these points exist on closed intervals, making it possible to solve real-world problems like maximizing profit or optimizing designs.

The method is a powerful tool that puts the Extreme Value Theorem into action. By checking and endpoints, we can pinpoint exactly where a function reaches its peaks and valleys, giving us crucial insights for practical applications.

The Closed Interval Method and the Extreme Value Theorem

Extreme Value Theorem implications

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  • States ff on closed interval [a,b][a, b] attains value f(c)f(c) and value f(d)f(d) at some numbers cc and dd in [a,b][a, b]
  • Guarantees existence of absolute extrema for continuous functions on closed intervals (sine function)
  • Allows use of closed interval method to find absolute extrema
  • Provides foundation for solving involving continuous functions on closed intervals (maximizing profit)

Steps of closed interval method

  1. Ensure function ff is continuous on closed interval [a,b][a, b]
  2. Find all of ff in interval [a,b][a, b]
    • Critical numbers are values of xx where f′(x)=0f'(x) = 0 or f′(x)f'(x) does not exist (cusps, discontinuities)
  3. Evaluate ff at each critical number found in Step 2
  4. Evaluate ff at endpoints of interval, aa and bb
  5. Compare all function values obtained in Steps 3 and 4
    • Largest value is absolute maximum (global maximum)
    • Smallest value is absolute minimum (global minimum)

Absolute extrema on closed intervals

  • Given continuous function ff on closed interval [a,b][a, b], follow steps of closed interval method
    • Find all critical numbers of ff in [a,b][a, b]
    • Evaluate ff at each critical number and endpoints aa and bb
    • Compare function values to determine absolute maximum and minimum
  • Example: Find absolute extrema of f(x)=x3−3x2−9x+1f(x) = x^3 - 3x^2 - 9x + 1 on interval [−2,4][-2, 4]
    • f′(x)=3x2−6x−9=3(x2−2x−3)=3(x−3)(x+1)f'(x) = 3x^2 - 6x - 9 = 3(x^2 - 2x - 3) = 3(x - 3)(x + 1)
    • Critical numbers: x=−1,3x = -1, 3
    • Evaluate ff at critical numbers and endpoints: f(−2),f(−1),f(3),f(4)f(-2), f(-1), f(3), f(4)
    • Compare values to find absolute maximum and minimum

Optimization with closed interval method

  • Optimization problems involve finding maximum or minimum value of function subject to constraints
  • Steps to solve optimization problems using closed interval method
    1. Identify objective function (function to be maximized or minimized)
    2. Determine constraints on variables (domain restrictions)
    3. Express objective function in terms of single variable
    4. Find closed interval over which objective function is defined
    5. Apply closed interval method to find absolute extrema of objective function on closed interval
    6. Interpret results in context of original problem
  • Example: Rectangular garden has perimeter of 200 ft. Find dimensions that maximize area.
    • Objective function: A(x)=x(100−x)A(x) = x(100 - x), where xx is width and 100−x100 - x is length
    • Constraints: 0<x<1000 < x < 100
    • Closed interval: [0,100][0, 100]
    • Apply closed interval method to find maximum area

Key Terms to Review (17)

Absolute maximum: An absolute maximum is the highest value of a function over its entire domain or within a specified interval. This term is crucial in finding points where a function reaches its peak, which can be identified through evaluating the function at critical points and endpoints. Understanding absolute maxima helps to analyze the behavior of functions and is connected to relative extrema and the closed interval method.
Absolute minimum: An absolute minimum is the lowest value of a function over its entire domain or within a specified interval. It represents the smallest output value that a function can achieve, which can occur at specific points or endpoints in the domain. Identifying absolute minima is crucial for understanding the overall behavior of functions and is particularly significant when considering both closed intervals and differentiable functions.
Boundedness: Boundedness refers to the property of a function where its output values remain confined within a specific range, meaning that there exists a real number that serves as both an upper and a lower limit for those values. This concept is crucial when discussing the behavior of functions, particularly in relation to continuity, limits, and optimization, as it helps determine whether functions exhibit certain characteristics over given intervals.
Closed Interval: A closed interval is a range of numbers that includes both its endpoints, denoted as $$[a, b]$$, where $$a$$ and $$b$$ are the minimum and maximum values respectively. This concept is crucial when discussing properties of continuous functions, as well as theorems that depend on the behavior of functions within specified limits. A closed interval ensures that both the starting point and the ending point are included in any calculations or conclusions drawn from the analysis of a function over that interval.
Continuity: Continuity in mathematics refers to a property of a function where it does not have any breaks, jumps, or holes over its domain. A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. This concept is crucial because it ensures that the behavior of functions can be analyzed smoothly, impacting several important mathematical principles and theorems.
Continuous Function: A continuous function is a function where small changes in the input lead to small changes in the output, meaning there are no breaks, jumps, or holes in the graph. This property is crucial for understanding various concepts in calculus, including limits, derivatives, and integrals, as it allows for the application of many fundamental theorems and methods without interruptions.
Critical Numbers: Critical numbers are values of a function's variable where the derivative is either zero or undefined. These numbers are important because they help identify potential local maxima and minima, as well as points of inflection. Understanding critical numbers is essential for analyzing the behavior of functions, especially when using methods like the closed interval method to find absolute extrema.
Critical Points: Critical points are specific values of a function's independent variable where the derivative is either zero or undefined. These points are important because they are potential locations for local maxima, local minima, or points of inflection, and play a crucial role in determining the behavior of a function in various contexts.
Differentiable function: A differentiable function is one that has a defined derivative at every point in its domain, indicating that it is smooth and continuous without any abrupt changes or breaks. This concept is vital as it relates to the behavior of functions, allowing us to analyze their rates of change and apply various calculus principles.
Endpoint evaluation: Endpoint evaluation is the process of assessing the values of a function at the endpoints of a closed interval, which are crucial for determining maximum and minimum values in that interval. This evaluation is significant because, in the closed interval method, it allows for identifying global extrema along with any critical points that exist within the interval. By considering both endpoints and critical points, a complete analysis of the function's behavior can be achieved.
Extreme Value Theorem: The Extreme Value Theorem states that if a function is continuous on a closed interval, then it must have both a maximum and a minimum value on that interval. This theorem is crucial because it connects the concepts of continuity, derivatives, and optimization, providing a foundation for finding absolute and relative extrema of functions.
F(a): In mathematics, f(a) represents the value of a function f at a specific input 'a'. This notation is crucial as it allows us to evaluate the function for different values, which is essential when applying methods to find maximum and minimum values within a closed interval. Understanding f(a) helps in determining how functions behave at particular points, which is fundamental in optimization problems.
F(b): In calculus, f(b) represents the value of a function f at a specific point b. It is essential for evaluating functions within a closed interval and is used to determine the function's behavior at the boundaries of that interval. Understanding f(b) helps in finding maximum and minimum values of a function, particularly when applied in optimization problems involving closed intervals.
Finding local maxima: Finding local maxima involves determining the highest points in a specific interval of a function, where the function value is greater than that of its neighboring points. This concept is crucial when analyzing the behavior of functions, as local maxima indicate peaks within given ranges and help identify optimal solutions in various applications, such as economics and engineering.
Finding local minima: Finding local minima refers to the process of identifying points in a function where the value is lower than all nearby points, essentially marking a 'valley' in the graph. This concept is crucial when analyzing functions on closed intervals, as it helps to determine the optimal solutions for maximizing or minimizing values. Local minima can occur at endpoints or critical points and are essential in understanding the behavior of functions within defined limits.
Function behavior: Function behavior refers to the way in which a function acts or changes over its domain, particularly how it behaves at critical points and intervals. Understanding function behavior helps identify features like increasing and decreasing intervals, local maxima and minima, and points of inflection. This analysis is essential for evaluating functions within a closed interval, as it provides insights into the function's overall shape and trends.
Optimization problems: Optimization problems involve finding the best solution or outcome from a set of possible choices, typically subject to certain constraints. These problems often require determining maximum or minimum values of a function, and are closely linked to concepts like rates of change, the application of derivatives, and methods for evaluating functions over defined intervals.
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