Absolute extrema on a closed interval

Problem 3.244 · hard

1. A continuous function on a closed interval has its extreme values at critical points or endpoints.
   Reviewed
2. \[ \frac{d}{d x} \left(- x^{3} + \frac{15 x^{2}}{2} - 18 x + 2\right) = \left(9 - 3 x\right) \left(x - 2\right) \]
   Differentiate and factor.✓ Proved
3. Critical numbers inside [-4, 4]: 2, 3.
   Reviewed
4. \[ \left. - x^{3} + \frac{15 x^{2}}{2} - 18 x + 2 \right|_{\substack{ x=-4 }} = 258 \]
   f(-4).✓ Proved
5. \[ \left. - x^{3} + \frac{15 x^{2}}{2} - 18 x + 2 \right|_{\substack{ x=2 }} = -12 \]
   f(2).✓ Proved
6. \[ \left. - x^{3} + \frac{15 x^{2}}{2} - 18 x + 2 \right|_{\substack{ x=3 }} = - \frac{23}{2} \]
   f(3).✓ Proved
7. \[ \left. - x^{3} + \frac{15 x^{2}}{2} - 18 x + 2 \right|_{\substack{ x=4 }} = -14 \]
   f(4).✓ Proved
8. The largest value is 258 and the smallest is -14.
   Reviewed