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Unformatted text preview: ould get some sort of consolation prize for being ‘the top of the hill’
between x = −4 and x = 3. We say that the function f has a local maximum7 at the point
(−2, 4.5), because the y -coordinate 4.5 is the largest y -value (hence, function value) on the curve
‘near’8 x = −2. Similarly, we say that the function f has a local minimum9 at the point (3, −8),
since the y -coordinate −8 is the smallest function value near x = 3. Although it is tempting to
say that local extrema10 occur when the function changes from increasing to decreasing or vice
versa, it is not a precise enough way to deﬁne the concepts for the needs of Calculus. At the risk of
being pedantic, we will present the traditional deﬁnitions and thoroughly vet the pathologies they
induce in the exercises. We have one last observation to make before we proceed to the algebraic
deﬁnitions and look at a fairly tame, yet helpful, example.
If we look at the entire graph, we see the largest y value (hence the largest function value) is 5.5
at x = 6. In this cas...
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