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**Unformatted text preview: **ts from Geometry have we not yet described algebraically?
What topics from Intermediate Algebra have we not yet discussed geometrically? 1.7 Graphs of Functions 79 10. It’s now time to “thoroughly vet the pathologies induced” by the precise deﬁnitions of local
maximum and local minimum. We’ll do this by providing you and your classmates a series
of exercises to discuss. You will need to refer back to Deﬁnition 1.8 (Increasing, Decreasing
and Constant) and Deﬁnition 1.9 (Maximum and Minimum) during the discussion.
(a) Consider the graph of the function f given below.
y
3
2
1
−2 −1
−1 1 2 x −2
−3 i.
ii.
iii.
iv. Show that f has a local maximum but not a local minimum at the point (−1, 1).
Show that f has a local minimum but not a local maximum at the point (1, 1).
Show that f has a local maximum AND a local minimum at the point (0, 1).
Show that f is constant on the interval [−1, 1] and thus has both a local maximum
AND a local minimum at every point (x, f (x)) where −1 < x < 1. (b) Using Example 1.7.4 as a guide, show that the function g whose graph is given below
does not have a local maximum at (−3, 5) nor does it have a local minimum at (3, −3).
Find its extrema, both local and absolute...

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