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5-1 POW! extreme values

Learn about the concept of extreme values in functions, including absolute, local, and relative extrema. Discover how to find and analyze these values using critical points, endpoints, and derivatives.

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5-1 POW! extreme values

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  1. 5-1 POW! extreme values Extreme Values of functions Today’s Date: 10/14/16 Test Date: Fri 11/4

  2. Def: (Absolute) Extreme Value If f is a function with Domain D, f (c) is a) the absolute maximum value on D ifff (x)  f (c) b) the absolute minimum value on D ifff (x)  f (c) Note: if the domain changes, so can the abs. max & min Thm: Extreme Value Theorem If f is continuous on a closed interval [a, b], then f has both a max value & a min value on the interval. Note: could be @ an endpoint or inside the interval

  3. Def: Local / Relative Extreme Values abs max local max local min local min abs min If c is an interior point in the domain of f , then f (c) is: a) a local max value ifff (x)  f (c) in an open interval b) a local min value ifff (x)  f (c) in an open interval Note: the list of local extrema will automatically include the absolute extrema

  4. Thm: Local Extreme Values A function has a local max or local min where f'(x) = 0 OR f'(x) DNE What do I check? f'(x) = 0 (2) f'(x) DNE (3) endpoints (if it’s closed) Def: Critical Point a point in the interior of a domain where f'(x) = 0 or f'(x) DNE Def: Stationary Point a point in the interior of a domain where f'(x) = 0 Note: Critical & stationary points do not have to be the same

  5. Ex 3) Find the absolute maximum and minimum values of on the interval [–2, 3]. • 0 • DNE @ x = 0 Critical value @ x = 0  f (0) = 0 Endpoints: @ x = –2  f (–2) = @ x = 3  f (3) = abs min @ (0, 0) abs max @

  6. happens @ Extreme Value Critical Point or Endpoint Critical Point or Endpoint Extreme Value does not have to be Ex 5) Find the extreme values of –4x = 0  x = 0 deriv DNE @ x = 1 no endpoints local max local min Critical values: @ x = 0  f (0) = 5 @ x = 1  f (1) = 3 Reminder: Solve analytically first THEN confirm w/ calculator

  7. homework Pg. 198 #3–8, 11, 14, 15, 19, 23, 29, 35, 38, 41

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