Absolute extrema on a closed interval

Problem 3.363 · hard

1. A continuous function on a closed interval has its extreme values at critical points or endpoints.
   Reviewed
2. \[ \frac{d}{d x} \left(- x^{3} - \frac{3 x^{2}}{2} + 18 x - 2\right) = \left(6 - 3 x\right) \left(x + 3\right) \]
   Differentiate and factor.✓ Proved
3. Critical numbers inside [-3, 3]: 2.
   Reviewed
4. \[ \left. - x^{3} - \frac{3 x^{2}}{2} + 18 x - 2 \right|_{\substack{ x=-3 }} = - \frac{85}{2} \]
   f(-3).✓ Proved
5. \[ \left. - x^{3} - \frac{3 x^{2}}{2} + 18 x - 2 \right|_{\substack{ x=2 }} = 20 \]
   f(2).✓ Proved
6. \[ \left. - x^{3} - \frac{3 x^{2}}{2} + 18 x - 2 \right|_{\substack{ x=3 }} = \frac{23}{2} \]
   f(3).✓ Proved
7. The largest value is 20 and the smallest is -85/2.
   Reviewed