note8 - Math021 week 8 Example 8.1 Find all the relative...

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Math021, week 8Example 8.1Find all the relative maximum and minimum of the functionf(x) =xex-ex-x2.
ConcavityDefinition 8.2
Example 8.5Find the points of inflection off(x) =e-x2/2and hence sketchits graph.
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Theorem 8.6 (Second Derivative Test)Suppose thatcis a critical pointof a functionf.1. Iff00(c)>0,fhas a relative minimum atc.2. Iff00(c)<0,fhas a relative maximum atc.3. Iff00(c) = 0,fmay have a relative maximum, minimum or neither ofthem.proof:Iff00(c)>0,f00(x)>0 whenxis close toc.Thus,f0is increasing over aninterval includingc. But we know thatf0(c) = 0. Therefore,f0(x)0 whenxis a bit less thancandf0(x)0 whenxis a bit greater thanc.fhas a relativeminimum atc. We may deal with the remaining case in whichf00(c)<0 in asimilar way.Example 8.7Find all the relative maximum and minimum of the functionf(x) = 2x3+ 3x2-12x+ 7.

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