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# 11.10 - Section 4.1 Key concepts Absolute(or global vs...

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Section 4.1 : Key concepts? ..?.. Absolute (or global) vs. relative (or local) maxima and minima, critical numbers A function f with domain D has a global maximum at c if f ( c ) f ( x ) for all x in the domain of f . We call f ( c ) the maximum value of f on D . We say f has a local maximum at c if f ( c ) f ( x ) for all x in some suitably small neighborhood of c (note: this requires that f ( x ) is defined in some neighborhood of c , so that in particular c cannot be an endpoint of D ). Global and local minima are defined in the same way, using instead of . Note that the function f ( x ) = x with domain [0,1] has no local maximum or minimum! It does however have a global maximum (at c =1) and a global minimum (at c =0).

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0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 The function f ( x ) = x with domain (0,1) has no global or local maxima or minima! Ditto for the function f ( x ) = x with domain R . The function f ( x ) = cos x with domain R has global maxima at all points c =2 π n ; it also has local maxima there.
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