5.2 Extreme Value Theorem, Global Versus Local Extrema

• The Extreme Value Theorem applies only to continuous functions on closed intervals.


  True
• Techniques for finding absolute extrema on closed intervals include checking critical points and endpoints
• What is one application of the Extreme Value Theorem in problem solving?

  Finding absolute extrema
• Global extrema are the absolute maximum and minimum values of a function over its entire domain.


  True
• Local extrema are the relative maximum and minimum values in a local region.
• The Extreme Value Theorem requires a function to be continuous on a closed interval to guarantee global extrema.


  True
• The Extreme Value Theorem requires that the function f(x) must be defined on a closed interval.
• Where can local extrema occur in relation to the derivative of a function?

  At critical points
• Match the type of extrema with its definition:

  Global Extrema ↔️ Absolute maximum/minimum over entire domain

  Local Extrema ↔️ Relative maximum/minimum in a local region
• What are the two primary methods for identifying critical points?

  f'(x) = 0 or undefined
• What are the critical points of f(x) = x^3 - 6x^2 + 5?

  x = 0, x = 4
• What is the definition of global extrema?

  Absolute max/min over domain
• What does the Extreme Value Theorem state?

  Global max and min exist
• What is the difference between global and local extrema?

  Absolute versus relative values
• Steps for finding local extrema using the first derivative test

  1️⃣ Find the derivative f'(x)

  2️⃣ Set f'(x) = 0 and solve for x

  3️⃣ Determine critical points

  4️⃣ Analyze the sign of f'(x) around critical points
• What interval condition is required for the Extreme Value Theorem?

  Closed interval
• Where may global extrema occur within the closed interval [a, b]?

  Critical points or endpoints
• Where may local extrema occur within the domain of a function?

  Anywhere in the domain
• What does the continuity condition ensure for the Extreme Value Theorem?

  No breaks in the graph
• Global extrema represent the absolute maximum and minimum values of a function over its entire domain.


  True
• What two conditions must be met for the Extreme Value Theorem to apply?

  Continuity and closed interval
• Critical points are values of x where the derivative f'(x) is either zero or undefined.
• Critical points occur where the derivative of a function is undefined because the function's slope may be vertical or have a cusp.


  True
• What two conditions must be met for the Extreme Value Theorem to apply?

  Continuity and closed interval
• Global extrema are located within the closed interval
• What does the Extreme Value Theorem state?

  Continuous functions have extrema
• The Extreme Value Theorem requires the function to be continuous on the entire closed interval
• Match the characteristic with the type of extrema:

  Definition ↔️ Absolute max/min over domain

  Location ↔️ Within closed interval
• Local extrema may occur anywhere in the domain.

  True
• What condition does the Extreme Value Theorem require for a function on a closed interval?

  Continuity
• What are the two methods for identifying critical points?

  Derivative equals zero or undefined
• Critical points are important for identifying both global and local extrema.


  True
• Critical points are important for identifying both global and local extrema
• Match the type of extrema with its definition:

  Global Extrema ↔️ Absolute max/min over entire domain

  Local Extrema ↔️ Relative max/min in a local region
• To find critical points, one method is to set the derivative equal to zero
• Steps for finding critical points using the derivative method:

  1️⃣ Find the derivative f'(x)

  2️⃣ Set f'(x) = 0

  3️⃣ Solve for x
• The Extreme Value Theorem applies to functions that are continuous on a closed interval.
• What is the derivative of f(x) = x^3 - 6x^2 + 5?

  3x^2 - 12x
• What are the critical points of f(x) = x^3 - 6x^2 + 5?

  0 and 4
• A local maximum occurs when f'(x) changes from positive to negative.


  True