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4.1 – Extreme Values of Functions

4.1 – Extreme Values of Functions. Absolute Extreme Values. Let f be a function with domain D. Then f(c) is the Absolute maximum value on D IFF f(x) ≤ f(c) for all x in D Absolute minimum value on D IFF f(x) ≥ f(c) for all x in D. Example 1.

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4.1 – Extreme Values of Functions

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  1. 4.1 – Extreme Values of Functions

  2. Absolute Extreme Values • Let f be a function with domain D. Then f(c) is the • Absolute maximum value on D IFF f(x) ≤ f(c) for all x in D • Absolute minimum value on D IFF f(x) ≥ f(c) for all x in D

  3. Example 1 • On [-π/2, π/2], f(x) = cosx takes on a maximum value of 1 (once) and a minimum value of 0 (twice). The function g(x) = sinx takes on a maximum value of 1 and a minimum value of -1.

  4. Example 2Absolute Extrema

  5. The Extreme Value Theorem

  6. Local Extreme Values Let c be an interior point of the domain of the function f. Then f(c) is a • Local maximum value at c IFF f(x) ≤ f(c) for all x in some open interval containing c. • Local minimum value at c IFF f(x) ≥ f(c) for all x in some open interval containing c. A function f has a local maximum or local minimum at an endpoint c if the appropriate inequality holds for all x in some half-open domain interval containing c. *Local extremaare also called relative extrema.

  7. Local Extreme Values Theorem • IF a function f has a local maximum value or a local minimum value at an interior point c of its domain, and if f’ exists at c, then f’(c) = 0

  8. Critical Point • A point in the interior of the domain of a function f at which f’ = 0 or f’ does not exist is a critical point of f.

  9. Example 3 • Find the absolute maximum and minimum values of f(x) = x2/3 on the interval [-2, 3]

  10. Example 4 • Find the extreme values of

  11. Example 5 • Find the extreme values of

  12. Example 6 Find the extreme values of

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