Absolute extrema on a closed interval

Problem 3.200 · hard

1. A continuous function on a closed interval has its extreme values at critical points or endpoints.
   Reviewed
2. \[ \frac{d}{d x} \left(x^{3} - \frac{15 x^{2}}{2} + 18 x\right) = \left(x - 2\right) \left(3 x - 9\right) \]
   Differentiate and factor.✓ Proved
3. Critical numbers inside [-3, 5]: 2, 3.
   Reviewed
4. \[ \left. x^{3} - \frac{15 x^{2}}{2} + 18 x \right|_{\substack{ x=-3 }} = - \frac{297}{2} \]
   f(-3).✓ Proved
5. \[ \left. x^{3} - \frac{15 x^{2}}{2} + 18 x \right|_{\substack{ x=2 }} = 14 \]
   f(2).✓ Proved
6. \[ \left. x^{3} - \frac{15 x^{2}}{2} + 18 x \right|_{\substack{ x=3 }} = \frac{27}{2} \]
   f(3).✓ Proved
7. \[ \left. x^{3} - \frac{15 x^{2}}{2} + 18 x \right|_{\substack{ x=5 }} = \frac{55}{2} \]
   f(5).✓ Proved
8. The largest value is 55/2 and the smallest is -297/2.
   Reviewed