Question: Given \(f(x)={{x}^{3}}-3{{x}^{2}}-9x+17\)

1. Find \(f'(x)\) and locate the first order critical numbers
2. Indicate the interval on which \(f\) is increasing and decreasing.

• The function is increasing if \(3(x-3)(x-1)>0\Leftrightarrow \left( x-3>0\text{ and }x-1>0 \right)\text{ or }\left( x-3<0\text{ and }x-1<0 \right)\)
  This means that \(x>3\text{ and }x>1\), or \(x<3\text{ and }x<-1\). Therefore, the function is increasing for \(x\in (-\infty ,-1)\cup (3,\infty )\)
• The function is decreasing if \(3(x-3)(x-1)<0\Leftrightarrow \left( x-3>0\text{ and }x-1<0 \right)\text{ or }\left( x-3<0\text{ and }x-1>0 \right)\)

This means that \(x>3\text{ and }x<1\), or \(x<3\text{ and }x>-1\). Therefore, the function is decreasing for \(x\in (-1,3)\)

c) Find both coordinates for the relative maximum and relative minimum points.

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