5.2 Extreme Value Theorem, Global Versus Local Extrema

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


  True
• The Extreme Value Theorem states that if a function f(x) is continuous on a closed interval [a, b], then f(x) must attain both a global maximum and a global minimum
• What does the Extreme Value Theorem ensure about continuous functions on closed intervals?

  Absolute extreme values exist
• How are global extrema defined in the context of the Extreme Value Theorem?

  Absolute max/min on interval
• What is the key difference between global and local extrema?

  Absolute vs relative values
• If the second derivative at a critical point is negative, the critical point is a local maximum
• Steps to identify local extrema:

  1️⃣ Find the critical points

  2️⃣ Evaluate the function at critical points

  3️⃣ Determine the type using the second derivative test
• What is the first condition for the Extreme Value Theorem?

  Function continuity
• Global extrema are absolute maximum and minimum values within a closed interval.


  True
• If the second derivative is positive at a critical point, it indicates a local minimum
• What is the purpose of the second derivative test in identifying local extrema?

  Determine the type of critical points
• The critical points of f(x) are the x-values where f'(x) = 0 or is undefined.


  True
• What does the Extreme Value Theorem state for a continuous function on a closed interval?

  Global max and min exist
• 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
• The interval required for the Extreme Value Theorem must be closed.


  True
• The Extreme Value Theorem requires the function to be differentiable on the closed interval.

  False
• To find local extrema, first find the critical points by setting the first derivative equal to zero
• Steps to identify local extrema using differentiation:

  1️⃣ Find critical points

  2️⃣ Evaluate the function at critical points

  3️⃣ Apply the second derivative test

  4️⃣ Classify each critical point
• A positive second derivative at a critical point indicates a local minimum.


  True
• The Extreme Value Theorem guarantees the existence of both a global maximum and a global minimum
• The interval for the Extreme Value Theorem must be closed
• Match the type of extrema with its description:

  Global Extrema ↔️ Absolute max/min on a closed interval

  Local Extrema ↔️ Relative max/min in a specific region
• If the second derivative is zero at a critical point, the second derivative test is inconclusive.


  True
• Steps to find local extrema using differentiation

  1️⃣ Calculate the first derivative f'(x)

  2️⃣ Set f'(x) = 0 and solve for x

  3️⃣ Determine where f'(x) is undefined

  4️⃣ Calculate the second derivative f''(x)

  5️⃣ Evaluate f''(x) at critical points

  6️⃣ Find the y-values of the extrema
• What type of extrema is x = 4 for f(x) = x³ - 6x² + 5?

  Local minimum
• If f''(0) = -6 for f(x) = x³ - 3x² + 2, then x = 0 is a local maximum
• The Extreme Value Theorem requires the function to be continuous on a closed interval.


  True
• The Extreme Value Theorem states that if a function f(x) is continuous on a closed interval [a, b], then f(x) must attain both a global maximum and a global minimum
• Why is the Extreme Value Theorem important in AP Calculus AB?

  Analyzes continuous function behavior
• Match the condition with its requirement for the Extreme Value Theorem:

  Function ↔️ Must be continuous on the closed interval

  Interval ↔️ Must be closed [a, b]
• Steps to identify global extrema using the Extreme Value Theorem:

  1️⃣ Find critical points

  2️⃣ Evaluate the function at critical points

  3️⃣ Evaluate the function at endpoints

  4️⃣ Identify the highest and lowest values
• If the second derivative at a critical point is positive, the point is a local minimum.


  True
• How do you find the critical points of a function?

  Set first derivative to zero
• What is the difference between local and global extrema?

  Relative vs absolute values
• The Extreme Value Theorem requires the function to be continuous on a closed interval.


  True
• How do you find global extrema on a closed interval?

  Critical points and endpoints
• What is the first step to identify local extrema of a function?

  Find critical points
• Steps to complete the identification of local extrema:

  1️⃣ Find critical points

  2️⃣ Evaluate the second derivative

  3️⃣ Find the y-values
• If f''(x) > 0 at a critical point, it is a local minimum
• If f''(x) < 0 at a critical point, it is a local maximum