02Relative Extrema - RELATIVE EXTREMA: FIRST AND SECOND...

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ELATIVE E XTREMA : F IRST AND S ECOND D ERIVATIVE T ESTS Theorem : Suppose that f is a function defined on an open interval containing the number 0 x . If f has a relative extremum at 0 x x = , then either 0 ) ( ' 0 = x f or f is not differentiable at 0 x . values of x where either 0 ) ( ' = x f or ) ( ' x f does not exist are called critical values when 0 ) ( ' = x f , we call these points stationary values (critical points) note: a critical point does not necessarily imply a relative extremum Consider the following graphs: ) ( ' x f is positive on the left, ) ( ' x f is negative on the left, same sign on both sides, negative on the right positive on the right no relative extrema This leads us to a theorem ) ( ' x f changes signs at relative extrema. First Derivative Test: Suppose f is continuous at a critical number 0 x . Absolute maximum Relative maximum Relative minimum There is an open interval 0 x on which ) ( 0 x f is the largest value. The relative maximum and
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02Relative Extrema - RELATIVE EXTREMA: FIRST AND SECOND...

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