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Assume that f is continuous on [-4,4] and differentiable on (-4,4). The table gives some values of f'(x) x: -4, -3, -2, -1, 0, 1, 2, 3, 4 f'(x): 74, 39, 10, -6, -14, -12, 0, 22, 55 a) estimate where f is increasing, decreasing, and has local extrema.

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f will be increasing on the intervals where
f'(x)>0 and decreasing wherever
f'(x)<0. Local extrema occur when
f'(x)=0 and the sign of
f'(x) changes to either side of that point.


f'(x) is positive when
x is between -4 and some number between -2 and -1, and also 2 (exclusive) and 4, so you can estimate that
f(x) is increasing on the intervals [-4, -2] and (2, 4].


f'(x) is negative when
x is between some number between -2 and -1, up to some number less than 2. So
f(x) is decreasing on the interval [-1, 1].

You then have two possible cases for extrema occurring. The sign of
f'(x) changes for some
x between -2 and -1, and again to either side of
x=2.
User David Jesse
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