5.2 Extreme Value Theorem, Global Versus Local Extrema

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

  Global max and min exist
• 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
• 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
• 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
• To find local extrema, first find the critical points by setting the first derivative equal to zero
• If the second derivative at a critical point is positive, the point is a local minimum.


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

  Critical points and endpoints
• 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
• What is the purpose of the second derivative test in identifying local extrema?

  Determine the type of critical points
• If f''(x) > 0 at a critical point, it is a local minimum
• If f''(x) = 0 at a critical point, the second derivative test is inconclusive.


  True
• Since f''(4) > 0, x = 4 is a local minimum
• Steps to find extrema using differentiation

  1️⃣ Find critical points by setting f'(x) = 0 or undefined

  2️⃣ Apply the first or second derivative test

  3️⃣ Determine the type of extremum (max/min)
• The critical points of f(x) = x³ - 3x² + 2 are x = 0 and x = 2
• What are the values of f(x) = x³ - 3x² + 2 at x = -1 and x = 3?

  f(-1) = -2 and f(3) = 2
• If a function f(x) is continuous on a closed interval [a, b], it must attain both a global maximum and a global minimum value within that interval.
• Local extrema must always exist within a closed interval.

  False
• The Extreme Value Theorem requires the interval to be closed.


  True
• The absolute maximum value of f(x) = x² - 4x + 3 on [0, 4] is 3, which occurs at x = 0 and x = 4.
• If f''(x) < 0 at a critical point, it indicates a local maximum.
• Global extrema are the absolute maximum and minimum values of a function over its entire domain.
• Global extrema refer to the absolute maximum and absolute minimum values of a function over its entire domain
• Global extrema are always the highest and lowest values over the entire domain
• What is the first step to find extrema using differentiation?

  Find the first derivative
• If f''(x) > 0 at a critical point, the point is a local maximum.

  False
• The local maximum of f(x) = x³ - 6x² + 5 is at (0, 5).


  True
• To find global extrema on a closed interval, you must evaluate the function at critical points and the endpoints
• What is the purpose of the first derivative test?

  Find local extrema
• Critical points must be included in an interval chart when applying the first derivative test.

  True
• The critical points of f(x) = x³ - 3x² + 2 are x = 0 and x = 2.


  True
• Steps to apply the first derivative test to find local extrema

  1️⃣ Find the first derivative f'(x)

  2️⃣ Determine critical points by solving f'(x) = 0

  3️⃣ Create an interval chart

  4️⃣ Test the sign of f'(x) in each sub-interval

  5️⃣ Find corresponding y-values
• If **f''(x) = 0** at a critical point, the second derivative test is inconclusive.


  True
• To apply the Extreme Value Theorem, evaluate f(x) at the critical points and the endpoints
• 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
• 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]