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4.3 Maxima and Minima

3 min readLast Updated on June 24, 2024

Maxima and minima are key concepts in calculus, helping us find the highest and lowest points of functions. These extrema can be absolute or local, and we use critical points to find them. Understanding these ideas is crucial for solving real-world problems.

The closed interval method is a powerful tool for finding extrema on a specific range. By evaluating critical points and endpoints, we can determine absolute maxima and minima. This technique has wide-ranging applications in fields like economics, physics, and engineering.

Maxima and Minima

Absolute vs local extrema

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  • Absolute extrema represent the highest (absolute maximum) or lowest (absolute minimum) points on the graph of a function within a given domain
  • Local extrema are points where the function value is greater than or equal to (local maximum) or less than or equal to (local minimum) the function values at nearby points
  • An absolute extremum is always a local extremum (global maximum or minimum) but a local extremum may not be an absolute extremum (local peak or valley)

Critical points in closed intervals

  • Critical points are points where the derivative of a function is either zero (f(x)=0f'(x) = 0) or undefined (corners, cusps, or discontinuities)
  • To find critical points, take the first derivative of the function, set it equal to zero and solve for x, and identify points where the derivative is undefined
  • A closed interval is a domain of a function that includes its endpoints, denoted as [a,b][a, b] (includes a and b)
  • The continuity of a function on a closed interval is crucial for applying the Extreme Value Theorem

Closed interval method for extrema

  1. Find the critical points of the function within the closed interval
  2. Evaluate the function at each critical point
  3. Evaluate the function at the endpoints of the closed interval
  4. Compare the function values from steps 2 and 3
  5. The largest value is the absolute maximum and the smallest value is the absolute minimum
  • The Extreme Value Theorem states that a continuous function on a closed interval must attain an absolute maximum and minimum value

Real-world significance of extrema

  • Maxima and minima can be used to solve real-world optimization problems
    • Maximizing profit, revenue, or efficiency (sales, production output)
    • Minimizing cost, time, or resource consumption (expenses, travel duration)
  • Extrema can help model and understand real-world phenomena
    • Maximum height of a projectile (cannon ball, rocket)
    • Minimum cost of production (manufacturing, construction)
    • Peak and trough points in economic cycles (stock market, business cycles)
  • Understanding the context and meaning of extrema in a given application is crucial for decision-making and problem-solving
  • Inflection points occur where the concavity of a function changes, often indicating a shift in the function's behavior
  • Saddle points are critical points that are neither local maxima nor local minima, resembling a saddle shape on the graph
  • Monotonicity refers to the property of a function being consistently increasing or decreasing over an interval, which can help identify potential extrema

Term 1 of 11

Absolute extremum
See definition

An absolute extremum is the highest or lowest value that a function attains on a given interval. It includes both absolute maximum and absolute minimum values.

Key Terms to Review (11)

Term 1 of 11

Absolute extremum
See definition

An absolute extremum is the highest or lowest value that a function attains on a given interval. It includes both absolute maximum and absolute minimum values.

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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

Term 1 of 11

Absolute extremum
See definition

An absolute extremum is the highest or lowest value that a function attains on a given interval. It includes both absolute maximum and absolute minimum values.



© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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