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Extreme Value Theorem in AP Calculus AB Day 1 Section 3.1

Learn about the Extreme Value Theorem in AP Calculus AB. Understand how a continuous function in the interval [a,b] must have an absolute maximum and minimum. Explore examples and types of extrema including absolute, relative, and local extrema.

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Extreme Value Theorem in AP Calculus AB Day 1 Section 3.1

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  1. AP Calculus AB Day 1 Section 3.1 Perkins

  2. Extreme Value Theorem (EVT) If f is continuous for all x in [a,b] then f must have both an absolute maximum and an absolute minimum value in the interval [a,b]. Examples: Absolute Maximum Abs Max Abs Max Abs Min Abs Min Absolute Minimum

  3. Extrema – highest and lowest function values when the derivative is either zero or undefined Horizontal tangents Vertical tangents 3 types Absolute (See previous slide) Relative -- occur only at critical numbers (cannot be at endpoints of an interval) ‘Local’ -- compared to the points we can see (can be at endpoints and critical numbers) A/R/L A/L A/L Local R/L Local A/L Local R/L A/R/L A/R/L Extrema will only occur at endpoints or critical numbers!

  4. Extrema occur at either endpoints or critical numbers. f(x) is continuous on [-1,3]. Minimum Find the extrema of on [-1,3]. Maximum Vertical tangents Horizontal tangents

  5. AP Calculus AB Day 1 Section 3.1 Perkins

  6. Extreme Value Theorem (EVT) If f is continuous for all x in [a,b] then f must have both an absolute maximum and an absolute minimum value in the interval [a,b]. Examples:

  7. Extrema 3 types Absolute Relative ‘Local’

  8. Find the extrema of on [-1,3].

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