5.2 Extreme Value Theorem, Global Versus Local Extrema

• What are extrema of a function on a given interval?

  Maximum and minimum values
• Absolute maximum is the largest value the function takes on the interval
• The absolute maximum of a function occurs at the highest point on its graph within the interval.
• What is a relative maximum of a function?

  Value greater than nearby points
• The Extreme Value Theorem guarantees that a continuous function on a closed interval will have both an absolute maximum and an absolute minimum
• What is the absolute maximum of $f(x) =$$x^{2}$ on $[ - 2, 2]$?


  $4$
• What is the absolute minimum of $f(x) =$$x^{2}$ on [ - 2, 2]</latex>?


  $0$
• Relative maximum and minimum exist for $f(x) =$$x^{2}$ on the closed interval $[ - 2, 2]$.


  False
• Local extrema are the maximum or minimum values within a specific interval
• What is a global extremum of a function?

  Absolute maximum or minimum
• Global extrema are always within a specific interval.

  False
• A local extremum occurs within a limited specific interval
• Where does a global extremum occur in relation to the domain of a function?

  Across the entire domain
• Global extrema are the highest or lowest values a function achieves across its entire domain.
• Order the types of extrema by their scope, from smallest to largest.

  1️⃣ Local Extrema

  2️⃣ Global Extrema
• What is the global maximum of f(x) = x^{2}</latex> on $[ - 2, 2]$?


  $4$
• What is the global minimum of $f(x) =$$x^{2}$ on $[ - 2, 2]$?


  $0$
• Extrema refer to the maximum and minimum values of a function
• Local extrema are the maximum or minimum values within a specific interval
• Global extrema are the absolute maximum or minimum values across the entire domain
• Match the type of extrema with its description:

  Local Extrema ↔️ Maximum or minimum within an interval

  Global Extrema ↔️ Absolute maximum or minimum
• Global extrema are the absolute maximum and minimum values of a function across its entire domain
• Local extrema are the maximum or minimum values of a function within a specific interval
• A local maximum occurs at a point where the function's value is greater than or equal to all nearby points
• Global extrema represent the absolute highest and lowest values across the entire domain of a function.
• The Extreme Value Theorem states that a continuous function on a closed interval must have both an absolute maximum and an absolute minimum
• Steps to apply the Extreme Value Theorem to find global extrema:

  1️⃣ Verify that the function is continuous on a closed interval

  2️⃣ Find the critical points by setting f'(x) = 0

  3️⃣ Evaluate f(x) at the critical points and endpoints

  4️⃣ Compare the values to determine global extrema
• Local extrema are the maximum or minimum values within a specific interval
• Global extrema are the absolute maximum or minimum values across the entire domain
• What are local extrema defined as?

  Maximum or minimum within an interval
• Global extrema represent the absolute maximum or minimum values across the entire domain of a function.
• Extrema refer to the maximum and minimum values of a function
• Match the type of extrema with its scope:

  Local Extrema ↔️ Specific interval

  Global Extrema ↔️ Entire domain
• Global extrema are the absolute maximum and absolute minimum values of a function across its entire domain.
• Local extrema can occur at the peaks and valleys of a function's graph.
• What is a local maximum?

  Value greater than nearby points
• Local extrema are categorized into local maximum and local minimum values within a specific interval.
• Match the type of local extrema with its characteristic:

  Local Maximum ↔️ Peaks in the graph

  Local Minimum ↔️ Valleys in the graph
• Local extrema are the maximum or minimum values of a function within a specific interval
• A local maximum is a value greater than or equal to all nearby points within a specific interval.