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3.1 Extrema

3.1 Extrema. Max/Min. Extreme Value Theorem. If f is continuous on [ a,b ] then f has both a minimum and a maximum on the interval If there is an open interval containing c on which f(c) is the maximum then f(c) is a relative maximum of f at ( c,f (c)).

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3.1 Extrema

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  1. 3.1 Extrema

  2. Max/Min

  3. Extreme Value Theorem • If f is continuous on [a,b] then f has both a minimum and a maximum on the interval • If there is an open interval containing c on which f(c) is the maximum then f(c) is a relative maximum of f at (c,f(c)). • If there is an open interval containing c on which f(c) is the minimum then f(c) is a relative minimum of f at (c,f(c)).

  4. Critical #s • Critical numbers: at the relative extrema the derivative = 0 or DNE • If f has a relative minimum/maximum at x=c then c is the critical # of f.

  5. Finding extrema • Find the extrema of

  6. Answer • Find critical #s by taking derivative and set = 0 • Plug in endpoints to get max/min

  7. Find extrema • Find the extrema of

  8. Answer • Take first derivative and set = 0.

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