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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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LammettHash

\(f\) 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 \(x=2\).

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